Logarithmic
LogarithmicFunctions
Functions
•
How do we write equivalent forms for exponential
and logarithmic functions?
•
How do we write, evaluate, and graph logarithmic
functions?
HoltMcDougal
Algebra 2Algebra 2
Holt
Logarithmic Functions
How many times would you have to double $1
before you had $512? You could solve this
problem if you could solve 2x = 512 by using
an inverse operation that undoes raising a
base to an exponent equation to model this
situation. This operation is called finding the
logarithm. A logarithm is the exponent to
which a specified base is raised to obtain a
given value.
Holt McDougal Algebra 2
Logarithmic Functions
You can write an exponential equation as a logarithmic
equation and vice versa.
Reading Math
Read logb a= x, as “the log base b of a is x.”
Notice that the log is the exponent.
Holt McDougal Algebra 2
Logarithmic Functions
Converting from Exponential to Logarithmic Form
Write each exponential equation in logarithmic form.
1.
2.
Exponential
Equation
Logarithmic
Form
35 = 243
log3243 = 5
1
2
25 = 5
log255 =
4
10
= 10,000
3.
4.
6–1 =
5.
ab = c
1
2
The base of the exponent becomes
the base of the logarithm.
The exponent is the logarithm.
log1010,000 = 4
1
6
Holt McDougal Algebra 2
log6
1
6
= –1
logac =b
An exponent (or log) can be negative.
The log (and the exponent) can be a
variable.
Logarithmic Functions
Converting from Exponential to Logarithmic Form
Write each exponential equation in logarithmic form.
Exponential
Equation
Logarithmic
Form
6.
9 = 81
log981 = 2
The base of the exponent becomes
the base of the logarithm.
7.
33 = 27
log327 = 3
The exponent of the logarithm.
logx1 = 0
The log (and the exponent) can
be a variable.
2
0
8. x = 1(x ≠ 0)
Holt McDougal Algebra 2
Logarithmic Functions
Converting from Logarithmic to Exponential Form
Write each logarithmic form in exponential equation.
Logarithmic
Form
Exponential
Equation
9.
log99 = 1
91 = 9
10.
log2512 = 9
29 = 512
11.
log82 =
12.
13.
log4
1
16
1
3
= –2
logb1 = 0
Holt McDougal Algebra 2
The base of the logarithm becomes
the base of the power.
The logarithm is the exponent.
1
3
8 =2
4–2 =
1
16
b0 = 1
A logarithm can be a negative
number.
Any nonzero base to the zero power
is 1.
Logarithmic Functions
Converting from Logarithmic to Exponential Form
Write each logarithmic form in exponential equation.
Logarithmic
Form
Exponential
Equation
14.
log1010 = 1
101 = 10
15.
log12144 = 2
122 = 144
16.
log 1 8 = –3
2
Holt McDougal Algebra 2
1
2
The base of the logarithm
becomes the base of the power.
The logarithm is the exponent.
–3
=8
An logarithm can be negative.
Logarithmic Functions
A logarithm with base 10 is called a common
logarithm. If no base is written for a logarithm,
the base is assumed to be 10. For example, log
5 = log105.
You can use mental math to evaluate some
logarithms.
Holt McDougal Algebra 2
Logarithmic Functions
Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
17. log 0.01 = x
1
10 = 0.01 
100
The log is the exponent.
10–2 = 0.01
Think: What power of 10 is 0.01?
x
log 0.01 = –2
Holt McDougal Algebra 2
Logarithmic Functions
Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
18. log5 125 = x
5x = 125
The log is the exponent.
53 = 125
Think: What power of 5 is 125?
log5125 = 3
Holt McDougal Algebra 2
Logarithmic Functions
Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
1
19. log5
5
x
5 =
=x
1
5
The log is the exponent.
1
5
5–1 =
1
log5 5
Think: What power of 5 is 1 ?
5
= –1
Holt McDougal Algebra 2
Logarithmic Functions
Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
20. log 0.00001 = x
10 x = 0.00001
The log is the exponent.
10–5 = 0.01
Think: What power of 10 is 0.01?
log 0.00001 = –5
Holt McDougal Algebra 2
Logarithmic Functions
Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
21. log25 0.04 = x
1
4

The log is the exponent.
25 = 0.04 
100 25
x
25–1 = 0.04
log250.04 = –1
Holt McDougal Algebra 2
Think: What power of 25 is 0.04?
Logarithmic Functions
Lesson 8.3 Practice A
Holt McDougal Algebra 2