7-3 Logarithmic Functions
Objectives
Write equivalent forms for exponential
and logarithmic functions.
Write, evaluate, and graph logarithmic
functions.
Holt Algebra 2
7-3 Logarithmic Functions
Vocabulary
logarithm
common logarithm
logarithmic function
Holt Algebra 2
7-3 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 Algebra 2
7-3 Logarithmic Functions
Example 1: Converting from Exponential to
Logarithmic Form
Write each exponential equation in logarithmic
form.
Exponential
Equation
Logarithmic
Form
35 = 243
log3243 = 5
1
2
25 = 5
log255 =
104 = 10,000
6–1 =
ab = c
Holt Algebra 2
1
6
1
2
The base of the exponent becomes
the base of the logarithm.
The exponent is the logarithm.
log1010,000 = 4
log6
1
6
= –1
logac =b
An exponent (or log) can be negative.
The log (and the exponent) can be a
variable.
7-3 Logarithmic Functions
Notes #1
Write each exponential equation in logarithmic
form.
Exponential
Equation
Logarithmic
Form
a.
9 = 81
log981 = 2
The base of the exponent becomes
the base of the logarithm.
b.
33 = 27
log327 = 3
The exponent of the logarithm.
logx1 = 0
The log (and the exponent) can
be a variable.
2
0
c. x = 1(x ≠ 0)
Holt Algebra 2
7-3 Logarithmic Functions
Example 2: Converting from Logarithmic to
Exponential Form
Write each logarithmic form in exponential
equation.
Logarithmic
Form
Exponential
Equation
The base of the logarithm becomes
the base of the power.
1
log99 = 1
9 =9
log2512 = 9
29 = 512
log82 =
log4
1
16
= –2
logb1 = 0
Holt Algebra 2
1
3
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.
7-3 Logarithmic Functions
Notes #2
Write each logarithmic form in exponential
equation.
Logarithmic
Form
Exponential
Equation
log101000 = 3
103 = 1000
log12144 = 2
log 1 8 = –3
2
Holt Algebra 2
122 = 144
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.
7-3 Logarithmic Functions
A logarithm is an exponent, so the rules for
exponents also apply to logarithms. You may have
noticed the following properties in the last example.
Holt Algebra 2
7-3 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 Algebra 2
7-3 Logarithmic Functions
Example 3A: Evaluating Logarithms by
Using Mental Math
Evaluate by without a calculator.
log 0.01
10? = 0.01
The log is the exponent.
10–2 = 0.01
Think: What power of 10 is 0.01?
log 0.01 = –2
Holt Algebra 2
7-3 Logarithmic Functions
Example 3B: Evaluating Logarithms by
Using Mental Math
Evaluate without a calculator.
log5 125
5? = 125
log5125 = 3
Holt Algebra 2
The log is the exponent.
7-3 Logarithmic Functions
Example 3C: Evaluating Logarithms by Using Mental
Math
Evaluate without a calculator.
log5 1
5
1
log5 5
Holt Algebra 2
= –1
7-3 Logarithmic Functions
Example 3D: Evaluating Using Mental Math
Evaluate without a calculator.
log250.04
log250.04 = –1
Holt Algebra 2
7-3 Logarithmic Functions
Because logarithms are the inverses of exponents, the
inverse of an exponential function, such as y = 2x, is a
logarithmic function, such as y = log2x.
You may notice that the
domain and range of each
function are switched.
The domain of y = 2x is all real
numbers (R), and the range is
{y|y > 0}. The domain of y =
log2x is {x|x > 0}, and the
range is all real numbers (R).
Holt Algebra 2
7-3 Logarithmic Functions
Example 4A: Graphing Logarithmic Functions
Use the x-values {–2, –1, 0, 1, 2}. Graph the
function and its inverse. Describe the domain
and range of the inverse function.
f(x) =
1
2
x
Graph f(x) = 12 x by
using a table of values.
x
f(x) =(
1
2
Holt Algebra 2
)
x
–2
–1
0
1
2
4
2
1
1
2
1
4
7-3 Logarithmic Functions
Example 4A Continued
To graph the inverse, f–1(x) =
log 1 x, by using a table of
2
values.
x
f
–1
(x) =log
1
2
x
4
2
1
–2
–1
0
1
2
1
1
4
2
The domain of f–1(x) is {x|x > 0}, and the range is R.
Holt Algebra 2
7-3 Logarithmic Functions
Notes (continued)
3A. Change 64 = 1296 to logarithmic formlog61296 = 4
B. Change log279 = 2 to exponential form. 27
3
Calculate the following using mental math
(without a calculator).
4. log864
2
5. log3 1
–3
27
Holt Algebra 2
2
3
=9
7-3 Logarithmic Functions
Notes (graphing)
6. Use the x-values {–1, 0, 1, 2} to graph
f(x) = 3x Then graph its inverse. Describe the
domain and range of the inverse function.
D: {x > 0}; R: all real numbers
Holt Algebra 2