How to Solve a Logarithmic Equation

Problem

\log_2 x = 5

Answer

x = 32

A logarithmic equation becomes a power equation once you use the definition. Raise the base to the value on the other side to recover the argument.

Step-by-step solution

Rewrite the logarithm as a power.
$\log_b x = y \Rightarrow b^y = x$
Why: A logarithm is an exponent, so the definition reverses the statement.
Apply the definition with base 2 and exponent 5.
$2^5 = x$
Why: The base of the log becomes the base of the power.
Evaluate the power.
$x = 2^5 = 32$
Why: Two to the fifth power is 32, and 32 is positive, so it is in the domain.

Multiplying 2 by 5 and writing x = 10. The log definition uses a power, not a product.
Swapping the base and the argument.
Forgetting that x must be positive. Logs only accept positive inputs.

Use the same method. Work each problem on paper, then open the answer to check.