Exponential And Logarithmic Equations Worksheet - Section 4-7 Page 7
ADVERTISEMENT
1
2
3
4
5
6
7
8
9
10
11320
4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS
E X A M P L E
Evaluating a Base 3 Logarithm
8
Evaluate log
5.2 to four decimal places.
3
Let y
log
5.2 and proceed as follows:
S o l u t i o n
3
log
5.2
y
3
y
5.2
3
Change to exponential form.
y
ln 5.2
ln 3
Take the natural log (or common log) of each side.
p
log
M
p log
M
y ln 3
b
b
ln 5.2
y
Solve for y.
ln 3
Replace y with log
5.2 from the first step, and use a calculator to evaluate the
3
right side:
ln 5.2
log
5.2
1.5007
3
ln 3
Evaluate log
0.0372 to four decimal places.
M A T C H E D P R O B L E M
0.5
8
To develop a change-of-base formula for arbitrary positive bases, with neither
base equal to 1, we proceed as above. Let y
log
N, where N and b are posi-
b
tive and b
1. Then
log
N
y
b
y
Write in exponential form.
N
b
y
Take the log of each side to another positive base
log
N
log
b
a
a
a, a
1.
p
log
M
p log
M
y log
b
b
b
a
log
N
a
y
Solve for y.
log
b
a
Replacing y with log
N from the first step, we obtain the chain-of-base formula:
b
log
N
a
log
N
b
log
b
a
In words, this formula states that the logarithm of a number to a given base is
the logarithm of that number to a new base divided by the logarithm of the old
base to the new base. In practice, we usually choose either e or 10 for the new
base so that a calculator can be used to evaluate the necessary logarithms (see
Example 8).
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education