Precalculus Exponential and Logarithmic Functions Solving Exponential and Logarithmic Equations
Solving Logarithmic Equations Concepts 1

Example 1. Solve log3(2x - 7) = 2.

Solution
The equation can be written equivalently as
2·x - 7 = 32.
2·x - 7 = 9.
2·x = 9 + 7.
2·x = 16.
x = 8.

Answer: x = 8.

Example 2. Solve log2(x - 5) + log2(x - 1) = 5.

Solution
We can write the sum of logarithms as one logarithm:
log2((x - 5)·(x - 1)) = 5.
The equation can be written equivalently as
(x - 5)·(x - 1) = 25.
(x - 5)·(x - 1) = 32.
x2 - 6·x - 27 = 0.
Using the quadratic formula to solve for x yields
x1 = 9; x2 = -3.
However, x2 = -3 must be ruled out since log2(x - 5) is not defined for x = -3.

Answer: x = 9 .

Example 3. Solve log3(x - 1) - log3(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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