Math Analysis – Precalculus, Sullivan 10th Edition
Ch5Sec4Day1 1 1/6/2020
Section 5.4 – Logarithmic Functions – Day 1
The logarithmic function with base a, where
0a
and
,1a
is denoted by
xlogy a
=
(read as “y is the logarithm with
base a of x”) and is defined by
xlogy a
=
if and only if
.ax y
=
The domain of the function
xlogy a
=
is
.0x|x
As this definition illustrates, a logarithm is a name for a certain exponent. So
a
log x
represents the exponent to
which a must be raised to obtain x. So, when you need to evaluate
a
log x,
think to yourself “a raised to what power
gives me x?”
Change Exponential Statements to Logarithmic Statements and Logarithmic Statements to Exponential
Statements
You should be able to change from exponential to logarithmic form and from logarithmic to exponential form.
Example 1: Change each exponential statement to Change each logarithmic statement to
an equivalent logarithmic statement. an equivalent exponential statement.
a) If
32a5=
, then
.532loga=
a) If
,x5log8=
then
.58x=
b) If
,x)4.6( 2=
then
.2xlog 4.6 =
b) If
,47logx=
then
.7x4=
Evaluate Logarithmic Expressions
To find the exact value of a logarithm, write the logarithm in exponential notation using the fact that
xlogy a
=
is
equivalent to
y
a x,=
and use the fact that if
,aa yx =
then
.yx =
Example 2: Find the exact value of
.81log3
To evaluate
3
log 81,
think “3 raised to what power yields 81?”
Then, let
81logy 3
=
.
The exponential form is
.813y=
4y 33 =
4y =
Thus,
.481log3=
Determine the Domain of a Logarithmic Function
The logarithmic function
xlogy a
=
has been defined as the inverse of the exponential function
.ay x
=
Thus, if
,a)x(f x
=
then
.xlog)x(f a
1=
−
Based on the discussion in Section 5.2, you know that for a function f and its inverse
,f 1−
Domain of f = Range of
1
f−
and Range of f = Domain of
1
f−
.
So, it follows that the Domain of the logarithmic function = Range of the exponential function
=
),0(
and the Range of the logarithmic function = Domain of the exponential function
=
).,( −
Summarizing some properties of the logarithmic function:
xlogy a
=
(defining equation:
y
ax =
)
Domain:
x0
Range:
− y
The domain of a logarithmic function consists of the positive real numbers, so the argument of a logarithmic function
must be greater than zero.
Example 3: Find the domain of each logarithmic function.
a)
)x2(log)x(f 4−=
Need
0x2 −
2x
Domain
2x|x =
or
)2,( −
