Download Worksheet 61 (11

Worksheet 61 (11.1)
Chapter 11
Exponential and Logarithmic
Functions
11.1 Exponents and Exponential Functions
The rules of exponents from chapter 5 can be extended to any real number
exponent.
If b > 0, b  1, and m and n are real numbers, then bn = bm if and only if
n = m.
Summary 1:
Warm-up 1. Solve:
1
81
-2x
(
3 =3
- 2x =
a) 3- 2x =
)
The solution set is {
x=
b)
}.
2 4 = 64
2 2x22 ( ) = 2( )
2x
x +1
2x
(
)
= 26
2 2
2x +
=6
x=
The solution set is {
262
}.
Worksheet 61 (11.1)
Problems - Solve:
1.
 21 2x - 1 = 321
2. 5 2x - 6 = 119
If b > 0 and b  1, then the function f defined by f(x) = bx, where x is any real
number, is called the exponential function with base b.
Summary 2:
Warm-up 2. Graph:
a) f(x) = 3x
x
f(x)
-2
1/9
Note: This is an example of an increasing function.
-1
0
1
2
f(x)
263
x
Worksheet 61 (11.1)
x
 1 
b) f(x) =   Note: This is an example of a decreasing function.
 3 
x
f(x)
-2
9
-1
0
1
2
f(x)
x
c) f(x) = 3x + 2
x
f(x)
-2
1
Note: This is a horizontal translation of f(x) = 3x.
-1
0
1
264
2
f(x)
x
Worksheet 61 (11.1)
x
 1 
d) f(x) =   + 2
 3 
Note:This is a vertical translation of f (x)=  31
x
f(x)
-2
11
x .
-1
0
1
2
f(x)
x
Problems - Graph:
 2 
3. f (x)=  
 5 
x
265
4. f ( x ) = 2 x - 3
Worksheet 62 (11.2)
11.2 Applications of Exponential Functions
Compound interest is an example of an exponential function.
General formula for compound interest:
nt
r 

A= P  1+  ;
n 

where P = principal, n = number of times being compounded,
t = number of years, r = rate of percent,
A = total amount of money accumulated.
Summary 1:
Warm-up 1. a) Find the total amount of money accumulated for $2000 invested at
12% compounded quarterly for 5 years.
r 

A= P  1+ 
n 

nt

(
A = 2000  1 +
(

(
A = 2000 (
)
)
)
(



)(
)
)
A=
The total amount of accumulation is __________.
Problem
1. Find the total amount of money accumulated for $1500 invested at 10%
compounded monthly for 3 years.
266
Worksheet 62 (11.2)
Summary 2:
n
1 

As n gets infinitely large, the expression  1 +  approaches the number
n 

e, where e equals 2.71828 to five decimal places.
The function defined by f(x) = ex is the natural exponential function.
Note: Use the ex key on the calculator to find functional values for x.
Formulas involving e:
1. A = P ert Used for compounding continuously.
A = total accumulated value, P = principal, t = years, r = rate
2. Q(t) = Q0 ekt Used for growth-and-decay applications.
Q(t) = quantity of substance at any time,
Q0 = initial quantity of substance, k = constant, t = time
267
Warm-up 2. a) The number of bacteria present in a certain culture after t hours is
given by the equation Q = Q0 e0.3t, where Q0 represents the number
of bacteria initially. If 18,149 bacteria are present after 6 hours, find
how many bacteria were present in the culture initially.
Q = Q0 e k t
= Q0 e( )( )
Q0 =
There were __________ bacteria initially.
Problem
2. Find the total accumulated money for $2000 invested at 12% compounded
continuously for 5 years.
Worksheet 63 (11.3)
11.3 Logarithms
If r is any positive real number, then the unique exponent t such that t = r
is called the logarithm of r with base b and is denoted by log b r.
Summary 1:
268
log b r = t is equivalent to bt = r.
For b > 0 and b  1, and r > 0,
1. log b b = 1
since b1 = b.
2. log b 1 = 0
since b0 = 1.
3. r = b log b r
since log b r = t.
Warm-up 1. Find the equivalent exponential expression:
a) log 5 125 = 3 is equivalent to 5 (
)
= 125.
b) log 10 10000 = 4 is equivalent to 10 (
c) log 2 32 = 5 is equivalent to 2 (
)
)
= 10000.
= 32.
Warm-up 2. Find the equivalent logarithmic expression:
a) 10-3 = 0.001 is equivalent to log 10 0.001 = _____.
b)
1
 
3
4
= 811 is equivalent to log 1
3
1
81
=
.
c) 54 = 625 is equivalent to log 5 625 = _____.
Problems
1. Find the exponential expressions for log 3 27 = 3 and log 10 .00001 = -5.
269
2. Find the logarithmic expressions for 102 = 100 and  41
3 = 641 .
Worksheet 63 (11.3)
Warm-up 3. a) Evaluate log 3 243 by first rewriting in exponential form and then
solving. (See summary 1 in section 10.1.)
Let log 3 243 = x
This is equivalent to: 3x = 243
3x = 3 ( )
x = _____
Therefore, log 3 243 = _____.
log 32 x = 52
b) Solve:
(

32
5
32
)

=x
(
)
=x
_____ = x The solution set is {
}.
Problems
3. Evaluate log 10 10000 by first rewriting in exponential form and then solving.
4. Solve:
log 125 x = 23
For positive real numbers b, r, and s where b  1,
log b rs = log b r + log b s
Summary 2:
270
Warm-up 4. a) If log 10 2 = 0.3010 and log 10 7 = 0.8451, evaluate log 10 14.
log 10 14 = log 10 (2  _____ )
= log 10 2 + _______________
= __________
Worksheet 63 (11.3)
b) If log 2 7 = 2.8074 and log 2 5 = 2.3222, evaluate log 2 35.
log 2 35 = log 2 ( _____  _____ )
= ____________ + ____________
= __________
Problems
5. If log 10 3 = 0.4771 and log 10 5 = 0.6990, evaluate log 10 15.
6. If log 2 7 = 2.8074 and log 2 3 = 1.5850, evaluate log 2 21.
Summary 3:
r
Forlog
positive
real
lognumbers
b  s =
b r - logb,
b sr, and s where b  1,
Warm-up 5. a) If log 10 101 = 2.0043 and log 10 23 = 1.3617, evaluate
.
log 10  101
23
 = log 10 101 - __________
log 10  101
23
= __________
b) If log 8 5 = 0.7740, evaluate log 8  645  . (Recall: 82 = 64)
log 8  645  = log 8 64 - __________
= ______ - .7740
= __________
Problems
7. If log 10 3 = 0.4771 and log 10 5 = 0.6990, evaluate log 10  53  .
Worksheet 63 (11.3)
271
8. If log 2 5 = 2.3219, evaluate log 2 
5
32
.
For positive real numbers b, r, and p where b  1,
log b r p = p (log b r)
Summary 4:
1
Warm-up 6. a) If log 10 1995 = 3.2999, evaluate log 10 (1995 )2 .
1
log 10 (1995 )2 = ( ) log 10 1995
= __________
b) Express as a simpler logarithmic expression: log b 3
log b 3
x
y
(
= log b  xy 
x
y
)
) log b  xy 
=(
= 31 (
-
)
c) Solve: log 3 (2x - 1) + log 3 (x + 1) = 2
log 3 (
)(
)=2
32 = ( )(
)
9 = _______________
0 = _______________
x = _____ or x = _____
Note: Logarithms are only defined for positive numbers. Negative results are extraneous.
The solution set is {
Worksheet 63 (10.3)
272
}.
Problems
9. If log 10 5 = 0.6990, evaluate log 10 54.
10. Express as a simpler logarithmic expression: log b
x2
y
11. Solve: log 5 (4x + 1) - log 5 (x - 1) = 1
Worksheet 64 (11.4)
11.4 Logarithmic Functions
273
A function defined by an equation of the form f(x) = log b x, b > 0 and b  1 is
called a logarithmic function.
y = log b x is equivalent to x = by.
f(x) = bx and g(x) = log b x are inverse functions.
Summary 1:
Warm-up 1. a) Graph: y = log 3 x
Note: This is the inverse of y = 3x from warm-up 2a in section 10.1. Inverses are reflections of each
other through the line y = x.
b) Graph: f(x) = log 3 (x - 2)
Note: This is a horizontal translation 2 units right.
274
Worksheet 64 (11.4)
Problems
Note: See warm-up 2b in section 10.1.
1. Graph: f(x) = log 1 x
3
2. Graph: f(x) = 2 + log 1 x
3
Logarithms with a base of 10 are called common logarithms.
Summary 2:
275
log 10 x = log x
Note: Use log key on calculator to evaluate common logarithms.
f(x) = log x and g(x) = 10x are inverse functions.
Warm-up 2. Evaluate to four decimal places:
a) log 1.25 = __________
b) log 12.5 = __________
c) log 125 = __________
d) log 1250 = __________
Worksheet 64 (11.4)
Problems - Evaluate to four decimal places:
3. log 0.0243
4. log 0.243
5. log 2.43
6. log 24.3
Warm-up 3. Find x to five significant digits:
a) log x = 0.4150
Note: Use 10x key on calculator to find x.
)
x = 10(
x = __________
b) log x = 1.6135
276
)
x = 10(
x = __________
Problems - Find x to five significant digits:
7. log x = 0.0101
8. log x = -4.321
Natural logarithms are logarithms that have a base of e.
log e x = ln x
Note: Use ln key on calculator to evaluate natural logarithms.
f(x) = ln x and g(x) = ex are inverse functions.
Summary 3:
Warm-up 4. Evaluate to four decimal places:
a) ln 1.25 = __________
b) ln 12.5 = __________
c) ln 125 = __________
d) ln 1250 = __________
Worksheet 64 (11.4)
Problems - Evaluate to four decimal places:
9. ln 0.0243
10. ln 0.243
11. ln 2.43
12. ln 24.3
Warm-up 5. Find x to five significant digits:
277
a) ln x = 0.4150
Note: Use ex key on calculator to find x.
)
x = e(
x = __________
b) ln x = 1.6135
)
x = e(
x = __________
Problems - Find x to five significant digits:
13. ln x = 0.0101
14. ln x = -4.321
Worksheet 65 (10.5)
11.5 Exponential Equations, Logarithmic Equations, and
Problem Solving
Summary 1:
278
If x > 0, y > 0, and b  1, then x = y if and only if log b x = log b y.
Warm-up 1. Solve to the nearest hundredth:
a)
b)
10x = 5
log ( ) = log ( )
(
)log 10 = log 5
(
)
x=
(
)
x = __________
The solution set is {
}.
The solution set is {
}.
5x + 1 = 7
log (
) = log (
)
(
)log 5 = log (
)
(
)
x+1=
(
)
x = __________
c) ln (x + 2) = ln (x - 3) + ln 2
ln (x + 2) = ln [2( )]
x + 2 = ____________
x = _____
The solution set is {
Problems - Solve to the nearest hundredth:
1. ex + 1 = 40
2. 72x = 11
Worksheet 65 (11.5)
3. log (x - 1) + log (x - 4) = 1
279
}.
Warm-up 2. Use the compound interest formula A = P  1 + nr
logarithms to solve:
n t and
a) How long will it take $1000 to double itself if invested at 10%
interest compounded quarterly? (Round to tenths.)
A = P  1 + nr
(
n t

(
)
)= (
)  1 +
(
)

4t
2=(
)
log 2 =
log 1.025
t = _________
(



)t
It will take _____ years.
Problem - Use the formula A = Pert and natural logarithms to solve:
4. How long will it take $1000 to double itself at 10% interest when compounded
continuously? (Round to nearest tenth.)
Worksheet 65
(11.5)
The change-of-base formula for logarithms:
Summary 2:
280
log a r =
log b r
; where a, b, and r are positive real numbers
log b a
and a  1 and b  1.
Warm-up 3. Approximate to 3 decimal places:
a)
log 3 15
log (
)
log (
)
 __________
log 3 15 =
Note: Either common or natural logarithms can be used to approximate logarithms with bases
other than 10 or e.
b)
log 5 0.004
log (
)
log (
)
 __________
log 5 0.004 =
Problems - Approximate to 3 decimal places:
5. log 6 88
6. log 2 0.001
281