Precalculus
Exponential and Logarithmic Functions
Solving Exponential and Logarithmic Equations
Solving Logarithmic Equations
Concepts 1
Example 1.
Solve
log
3
(2x - 7) = 2.
Solution
The equation can be written equivalently as
2·x - 7 = 3
2
.
2·x - 7 = 9.
2·x = 9 + 7.
2·x = 16.
x = 8.
Answer: x = 8.
Example 2.
Solve
log
2
(x - 5) + log
2
(x - 1) = 5.
Solution
We can write the sum of logarithms as one logarithm:
log
2
((x - 5)·(x - 1)) = 5.
The equation can be written equivalently as
(x - 5)·(x - 1) = 2
5
.
(x - 5)·(x - 1) = 32.
x
2
- 6·x - 27 = 0.
Using the quadratic formula to solve for x yields
x
1
= 9; x
2
= -3.
However, x
2
= -3 must be ruled out since log
2
(x - 5) is not defined for x = -3.
Answer: x = 9 .
Example 3.
Solve
log
3
(x - 1) - log
3
(x - 73) = 2.
Solution
We can write the difference of logarithms as one logarithm:
The equation can be written equivalently as
x - 1 = 9·(x - 73).
x - 1 = 9·x - 657.
x - 9·x = 1 - 657.
-8·x = -656.
Answer: x = 82.
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