Download Find the inverse of the function f(x)=3x-5.

Logarithms
The “I’m going to lie to you a bit”
version
Example
• Every year I double my money
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
Year 0
$1
Year 1
$1
$1
Year 2
Year 3
Example
• If I know what time it is and want to know how much money I have
$1
t
a=1(2 )
a=# of $
t=# of yrs
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
Year 0
$1
Year 1
$1
$1
Year 2
Year 3
Example
• What if I know money and want to know time?
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
Year 0
$1
Year 1
$1
$1
Year 2
Year 3
Example
• How long would it take me to get $1,000,000?
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
Year 0
$1
Year 1
$1
$1
Year 2
Year 3
Two sequences
+1
+1
+1
+1
+1
+1
+1
+1
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
2
4
8
16
32
54
128
*2
*2
*2
*2
*2
*2
*2
*2
Exponential
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
2
4
8
16
32
54
128
*2
a=2t
Logarithmic
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
2
4
8
16
32
54
128
*2
t=log2a
Exponential
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
3
9
27
81
243
729
2187
*3
a=3t
Logarithmic
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
3
9
27
81
243
729
2187
*3
t=log3a
Exponential
The special base 10
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
10
100
1000
10000
100000
100000
10000000
*10
a=10t
Logarithmic
The special base 10
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
10
100
1000
10000
100000
100000
10000000
*10
t=log10a
t=log(a)
Exponential
The special base e
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
e
e2
e3
e4
e5
e6
e7
*e
a=et
Logarithmic
The special base e
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
e
e2
e3
e4
e5
e6
e7
*e
t=logea
t=ln(a)
Logarithmic
The special base e
e≈2.7182818284
This number makes calculus easier
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
e
e2
e3
e4
e5
e6
e7
*e
t=logea
t=ln(a)
Meanings
• a=2t
 I know t.
 a is the result I get from raising 2 to the t power.
• t=log2(a)
 I know a.
 t is the power I need to raise 2 to to get a.
Example
• a=23
 a is the result I get from raising 2 to the power 3.
 That result is 8. a=8.
• t=log2(8)
 t is the power I need to raise 2 to to get 8.
 Since 23=8, t=3.
Exponential
a=2t
Years (t)
-3
-2
1
0
1
2
3
4
Dollars (a)
1/8
¼
½
1
2
4
8
16
*2
No matter what power I use,
the result is always positive
Exponential
a=2t
Years (t)
-3
-2
1
0
1
2
3
4
Dollars (a)
1/8
¼
½
1
2
4
8
16
*2
There is no power that can get me a negative number
Logarithmic
t=log2a
Years (t)
-3
-2
-1
0
1
2
3
4
Dollars (a)
1/8
¼
½
1
2
4
8
16
*2
There is no power that can get me a negative number
Logarithmic
t=log2a
Years (t)
-3
-2
-1
0
1
2
3
4
Dollars (a)
1/8
¼
½
1
2
4
8
16
*2
I can only find the powers of positive numbers
Logarithmic
t=log2a
Years (t)
-3
-2
-1
0
1
2
3
4
Dollars (a)
1/8
¼
½
1
2
4
8
16
*2
I can only take the log of positive numbers
Logarithmic
t=log2a
Years (t)
-3
-2
-1
0
1
2
3
4
Dollars (a)
1/8
¼
½
1
2
4
8
16
*2
The domain of this function called “log2” is a>0
Logarithmic
t=log2a
Years (t)
-3
-2
-1
0
1
2
3
4
Dollars (a)
1/8
¼
½
1
2
4
8
16
*2
The domain of any log(whatever) is always whatever>0
Example problem
• Find the domain of 2log7(4x-3)+7x-9
Whatever is inside the log has to be >0.
I can find an answer whenever 4x-3>0
x>3/4
Find the domain:
a)
b)
c)
d)
e)
x > 5/4
x < 5/4
x > -5/4
x < -5/4
None of the above
Find the domain:
Whatever is inside the log has to be >0
5-4x>0
5>4x
5/4>x
b) x < 5/4
Rewriting equations
•
•
•
•
•
•
y=bx
2=3p
q+3=79
9=32x+1
7=e4
x+2=102x-1
x=logby
p=log32
9=log7(q+3)
2x+1=log39
4=ln(7)
or
2x-1=log(x+2) or
4=loge(7)
2x-1=log10(x+2)
• The result of the log is the exponent.
• The result of the exponent is what goes inside the
log.
Meanings
• a=2t
 I know t.
 a is the result I get from raising 2 to the t power.
• t=log2(a)
 I know a.
 t is the power I need to raise 2 to to get a.
 What is 2log2(x)?
 the result I get from raising 2 to the power I need
to raise 2 to to get x = x.
Meanings
• a=2t
 I know t.
 a is the result I get from raising 2 to the t power.
• t=log2(a)
 I know a.
 t is the power I need to raise 2 to to get a.
 What is log2(2x)?
 The power that I need to raise 2 to so that I get
the result of raising 2 to the x power. =x
Rewriting equations version 2
• 9=32x+1
• Taking the log of both sides.
 Log3(9)=Log3(32x+1)
 Log39=2x+1
• Exponentiating both sides
 3Log3(9)=32x+1
 9=32x+1
Rewriting equations version 2
• 3x-7=e2x+1
• Taking the log of both sides.
 ln(3x-1)=ln(e2x+1)
 ln(3x-7)=2x+1
• Exponentiating both sides
 eln(3x-7)=e2x+1
 3x-7=e2x+1
Convert the following logarithmic expression into
exponential form: y = ln(x+2).
a) ey = x+2
b) 10y = x+2
c) e(x+2) = y
d) 10 (x+2) = y
e) None of the above
Convert the following logarithmic expression into
exponential form: y = ln(x+2).
y = ln(x+2)
ey=eln(x+2)
ey=x+2
A
Properties of logs
Exponential Property
adding
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
2
4
8
16
32
54
128
multiplying
23+4=2324
Logarithmic Property
adding
Years (t)
0
1
2
3
4
5
6
7
Dollars (a)
1
2
4
8
16
32
54
128
multiplying
Log(8*16)=log(8)+log(16)
The basic property of logarithims
• Loga(bc)=logab+Logac
Example
•
•
•
•
•
Loga(b4)
=loga(bbbb)
=loga(b)+Loga(bbb)
=loga(b)+Loga(b) +Loga(b) +Loga(b)
=4Loga(b)
The basic properties of logarithims
• Loga(bc)=logab+Logac
• Loga(bn)=n*logab
Example
•
•
•
•
•
•
•
•
x=log832 what is x?
Rewrite as an exponential equation
8x=32
Take log2 of both sides
Log2(8x)=Log232
xLog2(8)=Log232
x=Log2(32)/Log2(8)
x=5/3
Change of base
•
•
•
•
•
•
•
•
x=logay what is x?
Rewrite as an exponential equation
ax=y
Take logc of both sides
Logc(ax)=Logcy
xLogc(a)=Logcy
x=Logc(y)/Logc(a)
logay=Logc(y)/Logc(a)
Change of base
• Using this rule on your calculator
• logay=Logc(y)/Logc(a)
If you’re looking for the logay use…
Log(y)÷Log(a)
Or
ln(y)÷ln(a)
The basic properties of logarithims
• Loga(bc)=logab+Logac
• Loga(bn)=n*logab
• Logab=logc(b)/logc(a)
 Side effect: you only ever need one log button on
your calculator.
 Logab=log(b)/log(a)
 Logab=ln(b)/ln(a)
Warning: Remember order of operations
WRONG
log(2ax)
=x*log(2a)
=x*[log(2)+log(a)]
=x log 2 + x log a
CORRECT
log(2ax)
=log(2(ax))
=log(2)+log(ax)
=log(2)+x*log(a)
What about division?
•
•
•
•
•
•
Loga(b/c)
=Loga(b(1/c))
=Loga(bc-1)
=Loga(b) + Loga(c-1)
=Loga(b) + -1*Loga(c)
=Loga(b) - Loga(c)
The advanced properties of logarithims
• Loga(bc)=logab+Logac
• Loga(bn)=n*logab
• Logab=logc(b)/logc(a)
 Side effect: you only ever need one log button on
your calculator.
 Logab=log(b)/log(a)
 Logab=ln(b)/ln(a)
 Loga(b/c)=logab-Logac
What about roots?
log a ( n b )
1/n
log a (b )
1
log a (b)
n
The advanced properties of logarithims
• Loga(bc)=logab+Logac
• Loga(bn)=n*logab
• Logab=logc(b)/logc(a)
 Side effect: you only ever need one log button on
your calculator.
 Logab=log(b)/log(a)
 Logab=ln(b)/ln(a)
 Loga(b/c)=logab-Logac
 Loga(n√b̅)=[logab]/n
Expand using the properties of logarithms:
log(x2y)
a)
b)
c)
d)
e)
log(x) + log(y)
log(x)-log(y)
2 log(x)-log(y)
2 log(x)+log(y)
None of the above
Expand using the properties of logarithms:
log(x2y)
log(x2y)
=log((x2)y)
=log(x2)+log(y)
=2log(x)+log(y)
D