Download The exponent of a number says how many times to use the number

The exponent of a number says how many times to use
the number in a multiplication.
In 82 the “2” says to use 8 twice in a multiplication,
So 82 = 8 8 = 64
Exponents are also called Powers or Indices.
In words: 82 could be called "8 to the power 2" or "8 to the second power", or simply "8
squared"
Definition
In general ,and n are positive integers.
n
=
n factors
Example: a7
a7 = a × a × a × a × a × a × a = aaaaaaa
Notice how I just wrote the letters together to mean multiply? We will do that a lot here.
Example: x6 = xxxxxx
The Key to the Laws
Writing all the letters down is the key to understanding the Laws
Example: x2x3 = (xx)(xxx) = xxxxx = x5
Which shows that x2x3 = x5, but more on that later!
So, when in doubt, just remember to write down all the letters (as many as the exponent tells
you to) and see if you can make sense of it.
Table is a summary of the Laws of Indices :
Laws
Example
In general, where m and n are integers.
1. a m  a n  a mn
am
 a mn
n
a
2. (a
3.
m
) n  a mn
(ab) m  a m  b m
m
4.
am
a

 
bm
b
1
a n  n
a
5. a 0  1
00 undefined
And the law about Fractional Exponents:
6.
(  0)
( b  0)
 0)
(  0)
EXERCISE
1.1
Simplify the following , giving you answers in positive indices only.
1. 52 5 54
= ___________________________________________________
2. (-2)3 (-2)5
= ___________________________________________________
3. 32n+5 3n-1 3-3n-2 = ___________________________________________________
4. (x3y5)(xy2)
= ___________________________________________________
5. (8a2b3c)(22ab5c6)(32a4bc5)
= ___________________________________________________
= ___________________________________________________
6. 52 5 54
= ___________________________________________________
7. a3n-4 ÷ a3n-7
= ___________________________________________________
= ___________________________________________________
8.
= ___________________________________________________
9.
10. 513 513
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
11.
= ___________________________________________________
= ___________________________________________________
12.
= ___________________________________________________
13.
= ___________________________________________________
14.
= ___________________________________________________
= ___________________________________________________
15.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
16.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
17.
18.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
19.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
20. [(4− 1 )− 2 ÷ (2− 2 )5 ]2 ÷ [(2− 2 ) ÷ (2− 2 )− 1 ]4
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
HOMEWORK
1.1
Simplify the following , giving you answers in positive indices only.
1.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
2.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
3.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
4.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
5.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
6.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
7.
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
= ___________________________________________________
8. Three of the following 365, 2104 and 752 Any amount is minimal.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
9. If > 0 and
then find the value of
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
10. If > 0 and
then find the value of
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
nth Roots
Definitions
For any real numbers a and b, and any positive integer n,
if an = b, then a is the nth root of b.
NOTE: Every positive real number has two real number square roots. The number 0 has just
one square root, 0 itself. Negative numbers do not have real number square roots.
Example
Not to be confused
Example 1. Find the value of the square root of 9 and
9
square root of 9
9
Example 2. Find the value of the square root of 16 and 16
square root of 16
16
Example 3. Find the value of the cube root -8 and 3 - 8
cube root -8
3
-8
Sarub ________________________________________________________________________
_________________________________________________________________________
Properties of the nth Roots
Properties of the nth Roots
1.  a   a when n a is a real number.
n
2.
n
n
3.
4.
n
an  a
an  a
when a <0 and n is odd number.
when a<0 and n is even number.
n
a n b  n ab
n
a n a

, b ≠0
b
b
n
Example
Example 4
Simplify each of the following.
(a)
= ____________________________________________________
(b)
= ____________________________________________________
(c)
= ____________________________________________________
= ____________________________________________________
(d)
= ____________________________________________________
EXERCISE
Evaluate each of the following.
1.2
1.
5.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
2. (2
6.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
3.
7.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
4.
8.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
;a > 0, b>0
= ________________________________
= ________________________________
= ________________________________
= ________________________________
HOMEWORK
Level 1.
Evaluate each of the following.
1.2
1.
2.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
3.
4.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
5.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
6.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
Level 2.
7. Let 32x = 4y = 6-2z find the value of
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Level 3.
8. Let 2x = 3y = 4z = 24k and
find the value of m
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
9. Let
find the value of xyz
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
The rank of the square root, multiplication and division
Concept
1. The rank of the square root must be the same.
2. If the rank of the square root not the same we need to make the same by
bring to the rank of the square root find the least common multiple.
3. Properties of the nth Roots
n
a n b  n ab
n
a n a

, b ≠0
b
b
n
Example 1
Simplify each of the following.
(a)
12
(b)
2 12
75
= ____________________________________________________
= ____________________________________________________
(c) (2 3 3 )(3 3 5 )( 3 2 ) = ____________________________________________________
(d) (3
3 )(5 3 2 )
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Formula
Difference of squares.
a2 – b2 = ___________________________________________________________
The sum and difference of cubes.
a3 + b3 = ___________________________________________________________
a3 - b3 = ___________________________________________________________
Complete the square.
(a + b)2 = __________________________________________________________
(a – b)2 = __________________________________________________________
Example 2
Find the value of the product for :
a) (
3  2 )( 3 - 2 ) = ____________________________________________________
b) (
5  2 )( 5 - 2 )= ____________________________________________________
c) (2+
3 )(2- 3 )
= ____________________________________________________
d) (5+
21 )(5- 21 )
= ____________________________________________________
e)
1
3 2
= ____________________________________________________
f)
1
5- 2
= ____________________________________________________
g)
52
5 -2
5 -2
52
= ____________________________________________________
= ____________________________________________________
= ____________________________________________________
= ____________________________________________________
EXERCISE
Evaluate each of the following.
1.
4.
5- 2
5 2
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
2 22 3
12  8 - 32
2.
3
3
16  4  1
3
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
5
9 3 6 3 4
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
5.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
3.
3
1.3
1
1
1
1


 ... 
1 2
2 3
3 4
8 9
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
6.
1
1
1
1


 ... 
5 7
7  3 3  11
47  7
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
HOMEWORK
Evaluate each of the following.
Level 1.
1.
1.3
12
2 3 5
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Level 2.
1. If 2a =
Find the value of
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. What is a positive integer n.
1
1
1
1


 ... 
8
1 2
2 3
3 4
n  n 1
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Level 3.
3. If x =
,y=
Find the value of
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
To find the square root of the number in the form (a + b)

(a + b) + 2

the square root of (a + b) + 2
=
+2
+
=(
is
(
)2
+
+
)
but
and the square root of (a + b) - 2
=
but
Example 1
Find the square root of
a) 5 + 2
Concept
Trick
Concept
Trick
b) 4 -
2
Example 2
Find the value of square root.
1. 7  2 10 = ______________________
2. 4  2 3 =______________________
3. 22 - 2 105 =______________________
5.
6.
7.
4.
8. (6 
9.
8 - 2 15
=______________________
= _______________
= _______________
5 = ______________
9 - 80
4  15
30  10
1
35 ) 2
= ______________
8 - 45  52 - 6 35  17 - 12 2
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
10. If
4
17  288  a  b
Find the value of 2a + 3b
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
__________________________________________________________________________
Formula 2
n
a n a n a.................
1. Find the value of
8
= n-1 a
28 28 2......
Trick
Concept
Formula 3
a  a  a  ......
2. Find the value of
= 1
4a  1
2
6  6  6  ......
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Formula 4
a  a - a  ......
3. Find the value of
= 1
4a - 3
2
7  7 - 7  7 ......
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
EXERCISE
1.4
Find the square root of
1. 4 + 2
2. 16 - 2
3. 38 - 12
4. 12a + 2b - 4
5. 2x + 3 + 2
6.
Find the value of square root.
7.
8.
: a > 0 and b > 0
HOMEWORK
Level 1.
1.4
Find the value of
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Level 2.
Find the value of the fourth root 28 + 16
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Level 3.
Find the value of the cube root 26 + 15
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Solving equations involving square roots.
Concept
1. Isolate a square root.
2. Square both sides.
3. Repeat till all roots are gone.
4. Solve resulting equation.
5. Check solutions.
Example
Find the value of x
1.
=x
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. =
+1
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
3.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
4.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
EXERCISE
Find the value of x
1.
1.5
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. 3x2 – x + 5 +
= 12
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
HOMEWORK
Level 1.
Find the value of x
1.
1.5
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Level 2.
Find the value of x
1.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Level 3.
Find the value of x
1.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Exponential Function
Definition Exponential Function
An exponential function is a function in the form y =
, where a is a nonzero constant,
b is greater than 0 not equal to 1, and x is a real number.
f={(x,y)∈R × R+/y=ax;a>0 และ a  1}
Example
Concept
y = 0.5
f(x) =
Find the Exponential Function
1. y= (-3)x
2. y= ( 1 ) x
2
3. y= ( ) x
4. y=(0)x
5. y=(1)x
Increasing function and Decreasing function
Rule
Increasing Function
Increasing Function can be modeled with the function
y=
for a > 0 and b > 1
Rule
Decreasing Function
The function y =
models exponential decay for a > 0 and 0 < b < 1
Graph each function
1.
y=  1 
2
x
x -3 -2 -1 0
y
2. y=  1 
 3
1
2
3
3. y=2x
x -3 -2 -1 0
y
x
x -3 -2 -1 0
y
1
2
3
4. y=3x
1
2
3
x -3 -2 -1 0
y
1
2
3
Identify each function as increasing function or decreasing function.
1. y = 100
is ______________________________
2. y =
is ______________________________
3. y =
is ______________________________
4. y=
is ______________________________
5. y =
is ______________________________
6. y=( π )x
is ______________________________
7. y =
;a>0
8. y =
is ______________________________
Solve each equation
1.
3.
1
 
2
is ______________________________
x
= 64
2.
4. 4-x=
1
64
Solve each Inequality.
1.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
3.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
4.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
5.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
6.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
7.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
8.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Write an exponential function to each equation. Find domain and range to each equation.
1) y = 7-x
2) y = -(7-x)
y
y
x
0
Dr =___________________
Rr =___________________
3) y = -(7x)
Dr = ___________________
Rr = ___________________
4) y = 2x – 3
y
y
x
0
Dr =___________________
Rr =___________________
1
 
2
6) y =
y
y
0
Dr =___________________
Rr =___________________
x
0
Dr =___________________
Rr = ___________________
x
5) y = 4 + 2
x
0
x
( x 1)
1
0
Dr = ___________________
Rr = ___________________
x
7) y = -3x+1 + 2
y
8) y = 2 x
x
0
0
Dr =___________________
Rr =___________________
9. y =  1 
2
y
x
Dr = ___________________
Rr = ___________________
10. Math function with the graph of the function.
y
(0,3)
(0,2)
0
x
Dr = ___________________
Rr =___________________
x
0
y
x
a. y = 5-x + 3
b. y = -5x + 3
c. y = 5-x + 1
d. y = -5-x + 3
Solving equations involving exponential function
Form
Concept
1) ax=ay
2) ax=bx
3) ax2+bx+c=0
Example 1
Slove for x.
1.
2.
3.
4. (2x)x = 4(1-x)
Example 2
Slove for x.
1.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
3. If
and
then find the value of x + y .
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
HOMEWORK
2.1
Level 1.
Slove for x.
1.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Level 2.
3.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Level 3.
4.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
5.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Logarithmic function
Logarithms are the "opposite" of exponentials, just as subtraction is the opposite of addition
and division is the opposite of multiplication. Logs "undo" exponentials. Technically
speaking, logs are the inverses of exponentials.
In practical terms, I have found it useful to think of logs in terms of The Relationship
Definition Logarithmic Function
The logarithmic function with base b, where b > 0 and b 1, is denoted by
defined by
y=
if and only if
f = {(x,y) R+  R/ y= log x ; a>0 และ a  1}
a
Concept
and is
Find the logarithmic function.
1.
y  log 5 x
2.
y  log 1 x
3.
y  log 0 x
4.
y  log  x
5.
y  log 3 (5)
6. y  log
5
0
Rewriting in logarithmic form.
1. 24 = 16
__________________
4.
__________________
2. 102 = 100
__________________
5.
___________________
3.
__________________
6.
___________________
Rewriting in exponential form.
1.
__________________
4.
__________________
2.
__________________
5.
__________________
3.
__________________
6.
___________________
Increasing function and Decreasing function
Rule
Increasing Function
Increasing Function can be modeled with the function
y=
for x > 0 and b > 1
Rule
Decreasing Function
The function y =
models exponential decay for x > 0 and 0 < b < 1
Identify each function as increasing function or decreasing function.
1. y  log x
4. y = lnx
1
2
2.
3.
y  log 3 x
y  logx
5.
6.
y  log -2 x
Properties of the logarithmic function.
When a,M,N is a real number. a ≠1 and k is a real number.
1. log a 1 = 0
2. log a a = 1
3. log a MN = log a M + log a N
4. log a MN = log a M - log a N
5. alog M = M
6. log a M k = k log a M
7. log a M = 1k log a M
a
k
log a x
log a b
9. log b a  1
log a b
8. log
10.
b
x
EXERCISE
Evaluate each of the following.
3.1
1.
= ________________________________
= ________________________________
4.
= ________________________________
= ________________________________
2.
5.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
3.
6.
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
= ________________________________
7.
=__________________________________________________________________________
=__________________________________________________________________________
=__________________________________________________________________________
8.
= ________________________________
= ________________________________
= ________________________________
10.
9.
= ________________________________
= ________________________________
= ________________________________
=__________________________________________________________________________
=__________________________________________________________________________
=__________________________________________________________________________
11.
=__________________________________________________________________________
=__________________________________________________________________________
=__________________________________________________________________________
12. When a, b, c, d
and a, b, c, d 0 Find the value of
=__________________________________________________________________________
=__________________________________________________________________________
=__________________________________________________________________________
13.
= ________________________________
= ________________________________
15.
= ________________________________
= ________________________________
14.
= ________________________________
= ________________________________
16.
= ________________________________
= ________________________________
17.
=__________________________________________________________________________
=__________________________________________________________________________
=__________________________________________________________________________
=__________________________________________________________________________
=__________________________________________________________________________
=__________________________________________________________________________
HOMEWORK
Level 1.
3.1
Evaluate 3A when A =
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Level 2.
Find the value of
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Level 3.
If a, b, c and d are real numbers greater than one and
.
Find the value of
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Write an logarithmic function to each equation. Find domain and range to each equation
1) y = log3x
y
2) y = log 2 x
3 y
x
0
Dr =___________________
Rr =___________________
3) y = log2(x+2)
Dr = ___________________
Rr = ___________________
4) y = log3(x-2)
y
y
x
0
Dr =___________________
Rr =___________________
6) y = log3(x-1) - 2
y
Dr =___________________
Rr =___________________
x
0
Dr = ___________________
Rr = ___________________
5) y = log3x + 2
0
x
0
y
x
0
Dr =___________________
Rr = ___________________
x
Common logarithm
The common logarithm is the logarithm to base 10. The notation log x
When 10 is used as a base, it is not necessary to indicate it in writing logarithms. For example,
log 100 = 2
is understood to mean the same as
log 10 100 = 2
If the base is other than 10, it must be specified by the use of a subscript to the right and below
the abbreviation "log." As noted in the foregoing discussion of natural logarithms, the use of
the distinctive abbreviation "In" eliminates the need for a subscript when the base is e.
The value of log N can be written as N0 × 10n when 1 ≤ N0<10,n ∈I .
so
logN = log(N0 × 10n)
= logN0 + log10n
= logN0 + n
The integral part is called the Characteristic and the fractional or the decimal part is
called the Mantissa.
Example 1. Find the value of Characteristic and Mantissa of log N.
logN
log(N0 × 10n)
ค่า characteristic
1. log 218
log(2.18 × 102)
2
2. log 21.8
_______________ ___________________
3. log 2.18
_______________ ___________________
4. log 0.218
_______________ ___________________
5. log 0.00218 _______________ ___________________
6. log 87.96
_______________ ___________________
7. log 87960 _______________ ___________________
ค่า mantissa
log 2.18
_______________
_______________
_______________
_______________
_______________
_______________
Example 2. Given log 4.85=0.6857 Find the value of log N.
1. log 485 = ______________________________________________
2. log 0.485 = ______________________________________________
3. log 0.000485 = ______________________________________________
4. log 4850000 = ______________________________________________
Example 3. Given log 896 has value mantissa = 0.9523. Find the value of log N.
1. log 8.96
= __________________________________________
2. log 0.00896
= __________________________________________
3. log 0.0000896 = __________________________________________
Example 4. Find the value of log 3.457
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Example 5. Given log 1.15 = 0.0607 and log 1.16=0.0645. Find the value of log 1153.
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HOMEWORK
Level 1.
1. Find the value of log 144 – 2log3 + log 25 – log 4
3.2
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Level 2.
2. Given log 3 = 0.4771. Find the value of log 0.027 .
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Level 3.
3. Find the value of Characteristic and Mantissa of log 0.0013506.
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The antilogarithm (also called an antilog) is the inverse of the logarithm transform.
Since the logarithm (base 10) of 1000 equals 3, the antilogarithm of 3 is 1000.
Properties of the antilogarithm
1. Antilog a = x when log x = a
2. Antilog(log a) = a
Example 1. Find the value of Antilog.
1. Antilog(log3) = _____________________
3. Antilog(log18 –log9) = _______________
2. Antilog(2log7) = ________________
4. Antilog(loga+logb) = ________________
Example 2. Let Antilog 0.4082 = 2.56 . Find the value of N.
1. log N = 4.4082
2. log N = 0.4082 – 2
_____________________________
_____________________________
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_____________________________
_
3. log N = -2.5918
_____________________________
_____________________________
_____________________________
_____________________________
_
5. log N = -3.5918
_____________________________
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_
_____________________________
_____________________________
_____________________________
_____________________________
_
4. log N = -0.5918
_____________________________
_____________________________
_____________________________
_____________________________
_
6. log N = 8.4082 - 10
_____________________________
_____________________________
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_____________________________
_
The estimated values.
In some calculations involving multiplication, division, and the exponent in the form
of a hassle. We may come to the logarithm of the calculations. However, the calculated value
is an estimate only.
Example 1. Find the value of (0.653)(92.9)(214)
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Example 2. Find the value of
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Example 3. Find the value of
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EXERCISE
3.3
1. Let log 2 = 0.3010 and log N = -5 + 0.3010. Find the value of N.
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2. Calculate the number of how many 87524 numbers are assigned log2=0.3010,log7 = 0.8450
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3. Find that when the
will be zero after the decimal point.
How many decimal number. Let log2=0.3010.
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HOMEWORK
Level 1.
1. Find the value of
,
3.3
Let log 2.327 = 0.3667.
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Level 2.
2. Find the value of Antilog of 8log 2 – log 129 .
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Level 3.
Find the value of
by use logarithm .
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Any positive number is suitable as the base of logarithms, but two bases are used more
than any others:
base of
logarithms
symbol
name
10
log
(if no base shown)
common logarithm
e
ln
natural logarithm
Natural logs are logs, and follow all the same rules as any other logarithm. Just remember
lnx = logex =
log x
log x

loge
0.4343
Example
Evaluate each of the following.
1. ln 72, Let log 72 = 1.8573.
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2. ln 0.324 , Let Antilog 0.5105 = 3.24 .
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3. log[ln3.02+2ln3-ln10]10, Let loge = 0.4343,e=2.718
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EXERCISE
Evaluate each of the following.
1. ln 3 – ln 6 +ln 2
= ________________________________
= ________________________________
= ________________________________
3. ln 3470
= ________________________________
= ________________________________
= ________________________________
= ________________________________
3.4
2. eln 7
= ________________________________
= ________________________________
= ________________________________
4. ln 0.0753
= ________________________________
= ________________________________
= ________________________________
= ________________________________
5. e-ln2+ln10+ ln302 +ln0.1+ln0.09
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Solving Logarithmic Equations
The first type of logarithmic equation has two logs, each having the same base, set equal to
each other, and you solve by setting the insides (the "arguments") equal to each other. For
example:
Solve log2(x) = log2(14).
Since the logarithms on either side of the equation have the same base ("2", in this
case), then the only way these two logs can be equal is for their arguments to be equal. In
other words, the log expressions being equal says that the arguments must be equal, so I have:
x = 14
And that's the solution: x = 14
The second type of log equation requires the use of The Relationship:
Note that the base in both the exponential form of the equation and the logarithmic
form of the equation (above) is "b", but that the x and y switch sides when you switch between
the two equations. If you can remember this — that whatever had been the argument of the
log becomes the "equals" and whateverhad been the "equals" becomes the exponent in the
exponential, and vice versa — then you should not have too much trouble with solving log
equations.
Solve log2(x) = 4.
Since this is "log equals a number", rather than "log equals log", I can solve by using The
Relationship:
log2(x) = 4
24 = x
16 = x
Example.
Find the value of x.
1. logx = 4
2. log25x =
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3. log5(3x+2)=1
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5. log2(log3x)=2
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7. logx3
3
=
3
2
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9. logx2 = log x
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3
2
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4. log3(x2+2x)=1
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6. log4log3log2(x2-2x)=0
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8. logx 1 = - 23
8
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10. log3(
2
)=log3(4-x)
x -1
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EXERCISE
3.5
Find the value of x.
1. log5(x+2) = -log5x
2. log9x = log33x
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3. log3x – log3(2x+3)=-2
4. log(1+x)=1+ logx
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5. log5(x-1)+ log5(x-2)= log
5
6
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7. log(x-1)+log(x+1)=log(2x-1)
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9. log2x
log2 x
=1
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6.log5x + 2log5x =
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8.log16x+ log4x+ log2x=7
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10. log4log3log29 log
9
( x 2  2 x)
=0
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21. x log x
2
4
= x2-18x+34
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23.
logxlogx = 4
= 256
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27.
x log
=
x
2x
=4
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24. log3x=
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25. x log 4 x
22.
9
log 3 x
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26. x
3 log x
= 3 10,000
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28. logx=log52x
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29. 3log4x-2logx4 = 1
30. log5x + logx5 =
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HOMEWORK
3.5
Level 1.
Let log2 = 0.3010 ,log3= 0.4771 and log7 = 0.8451. Find the value of x.
1. 2x = 27
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_
3. 3x(2x) = 7
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_
2. 2x = 52x-1
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_
4. 32x-4(3x) +4 = 0
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_
Level 2.
5. Slove for x and y.
1. x + y = log79
2. x+y = log31
…………….(1)
…………….(2)
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6. 8x = 10y and 2x = 5y
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Level 3.
Given
. Find the value of x.
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Solving Inequalities Logarithms
Concept
1. If 0< a < 1 then
1.1
1.2
>
<
when x1 < x2
when x1 > x2
2. If a > 1 then
2.1
>
when x1 > x2
2.2
<
when x1 < x2
3. Behind the numbers of log. Must be a positive number.
4. The correct answer should be 1 and 2 and 3
Example.
Slove for x.
1. log4(2x+3)< log4(x-1)
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2. log 1 (2 - x) ≤ log 1 (
2
2
2
)
x 1
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EXERCISE
3.6
Slove for x.
1. log5(x2+3x-1) > log5(x-2)
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2.log4(x2+4x+11)<0
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3. log0.5(6+4x-x2) ≥ 0
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HOMEWORK
Level 1.
Slove for x.
1.
3.6
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Level 2.
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Level 3.
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