College Algebra 9th txtbk

358 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Z Properties of Logarithmic Functions
Some of the properties of exponential functions that we studied in Section 5-1 can be used
to develop corresponding properties of logarithmic functions. Several of these important properties
of logarithmic functions are listed in Theorem 2. We will justify them individually.
Z THEOREM 2 Properties of Logarithmic Functions
If b, M, and N are positive real numbers, b 1, and p and x are real numbers, then
1. log b 1 0
5. log b M log b N if and only if M N
2. log b b 1
6. log b MN log b M log b N
3. log b b x x
M
7. log b
N log b M log b N
4. b log b x x, x 7 0 8. log b M p p log b M
ZZZ CAUTION ZZZ
1. In properties 3 and 4, it’s essential that the base of the exponential and the base of
the logarithm are the same.
2. Properties 6 and 7 are often misinterpreted, so you should examine them carefully.
log b M
log b N log b M log b N
log b (M N) log b M log b N
log b M log b N log M b
N ;
log b M
cannot be simplified.
log b N
log b M log b N log b MN;
log b (M N) cannot be simplified.
Now we will justify properties in Theorem 2.
1. log because b 0 b 1 0
1.
2. log because b 1 b b 1
b.
3 and 4. These are simply another way to state that f (x) b x and f 1 (x) log b x are
inverse functions. Property 3 can be written as f 1 ( f (x)) x for all x in the domain
of f. Property 4 can be written as f ( f 1 (x)) x for all x in the domain of f 1 . This
matches our characterization of inverse functions in Theorem 5, Section 3-6. Together,
these properties say that if you apply an exponential function and a logarithmic function
with the same base consecutively (in either order) you end up with the same
value you started with.
5. This follows from the fact that logarithmic functions are one-to-one.
Properties 6, 7, and 8 are used often in manipulating logarithmic expressions. We will
justify them in Problems 111 and 112 in Exercises 5-3, and Problem 69 in the Chapter 5
Review Exercises.
EXAMPLE 5 Using Logarithmic Properties
Simplify, using the properties in Theorem 2.
(A) log e 1 (B) log 10 10 (C) log e e 2x1
(D) log 10 0.01 (E) 10 log 10 7
(F) e log e x 2