College Algebra 9th txtbk

SECTION 5–3 Logarithmic Functions 361
Z LOGARITHMIC–EXPONENTIAL RELATIONSHIPS
log x y
ln x y
is equivalent to x 10 y .
is equivalent to x e y .
EXAMPLE 8 Solving log b x y for x
Find x to three significant digits, given the indicated logarithms.
(A) log x 9.315 (B) ln x 2.386
SOLUTIONS (A) log x 9.315
Change to exponential form (Definition 1).
x 10 9.315
4.84 10 10
Notice that the answer is displayed in scientific notation in the calculator.
ln x 2.386
x e 2.386
10.9
(B) Change to exponential form (Definition 1).
MATCHED PROBLEM 8
Find x to four significant digits, given the indicated logarithms.
(A)
ln x 5.062
(B) log x 12.0821
ZZZ EXPLORE-DISCUSS 4
Example 8 was solved algebraically using logarithmic–exponential relationships.
Use the INTERSECT command on a graphing calculator to solve this problem
graphically. Discuss the relative merits of the two approaches.
Z The Change-of-Base Formula
How would you find the logarithm of a positive number to a base other than 10 or e? For
example, how would you find log 3 5.2? In Example 9 we evaluate this logarithm using several
properties of logarithms. Then we develop a change-of-base formula to find such logarithms
more easily.
EXAMPLE 9 Evaluating a Base 3 Logarithm
Evaluate log 3 5.2 to four decimal places.
SOLUTION Let y log 3 5.2 and proceed as follows:
log 3 5.2 y
5.2 3 y
ln 5.2 ln 3 y
ln 5.2 y ln 3 Solve for y.
ln 5.2
y
ln 3
Change to exponential form.
Apply the natural log (or common log) to each side.
Use log b M p p log b M, which brings the exponent y in front of ln 3 as a factor.