Exponential and Logarithmic Functions
Section 11-4
Logarithmic Functions
Definition
 The logarithmic function y = logax, where a>0
and a≠1,is the inverse ofthe exponential
function y = ax . So y=logax if and only if x = a y

Exponent
2 y = 32
Y=log2 32
Base
5
Since 2 = 32, y=5 Thus log 2 32 = 5
Exponential and logarithmic
are inverses of each other
Example # 1
 Write each equation in exponential form
1. log27 3 = 1/3
2. log16 4 = ½
1
3
Solution: 27 = 3
1
Solution: 16 2 = 4
Write each equation in logarithmic form
1 0
1. 2 = 1024
Solution: log2 1024 = 10
1
2. 2  3 = 1/8
Solution: log2 8 = - 3
Evaluating Logarithms
 Evaluate the expression log5
1
625
1
625
 Solution: Let x = log5
x
5 = 1
By definition of logarithms
x
625
5 = 1
5
4
x = -4
REMEMBER” Logarithmic Functions and Exponential Functions are
inverses of each other!!!!
 PROPERTIES OF LOGARITHMS
Solving logarithmic Equations
 Equations can be written involving logarithms. Use the properties of logarithms and
the definition of logarithms to solve these equations. Always check values if your
domain is not the set of all real numbers to determine if it is a valid solution. You can
not have a log of a negative number NOT Allowed log -7
 Example # 2: Solve each equation
 1
 1. logb 15625 =
3
1
6
b
 1
3
=
1
15625 6
 1
3
= 5
(b
 2. log 10 (2x+5) = log 10 (5x-4)
2x+5 = 5x-4
 3. Log 3 (4x+5)-log 3 (3-2x) = 2
log3 4x+5/3-2x =2
9 (3-2x) = 4x+5
27 -18x = 4x + 5
)
3
= 5
3
b=
x=3
22x = 22
3 2 = 4x+5/3-2x
x=1
1
125
Graphing logarithms
 You can graph a logarithm function by rewriting the log function as
an exponential function and constructing a table of values.
 Example Graph y = log 2 (x-1)
y
 Solution: Rewrite it as 2 = x-1
 Set up a table of values and
 Then plot the graph
Graphing a logarithmic inequality
First graph the boundary line. That is
graph the equality. Pay attention to
whether it is a dashed or solid line.
Next test a point on one side of the
curve and shade the side that makes
the inequality true.
HW # 28
Section 11-4
Pp. 723-725
#20-30 all, 33-47 odds, 53,67,68,71,75,80