Logarithm Power Rule
The logarithm of a power equals the exponent times the logarithm.
Formula
Variables
Example
log₂(8³) = 3·log₂ 8 = 3×3 = 9.
How it works
A power inside a logarithm can be brought out in front because logarithms measure repeated multiplication. If x is raised to the n, then x^n is n copies of x multiplied together, and the log of that product becomes n times the log of x by the product rule applied repeatedly. This rule only applies when the logarithm is defined, so the base b must be positive and not 1, and x must be positive.
When to use it
Use this rule when simplifying logarithmic expressions that contain exponents, or when solving equations where a variable appears inside a power under a log. It is also helpful in algebra, calculus, and scientific formulas where exponential growth or decay is rewritten in log form.
Worked example, step by step
Evaluate log_5(125^2) 125 = 5^3, so 125^2 = (5^3)^2 = 5^6 log_5(125^2) = log_5(5^6) = 6 · log_5(5) = 6 · 1 = 6
Common mistakes to avoid
- Forgetting that the exponent multiplies the log, not the base
- Using the rule when the logarithm argument is negative or zero
- Thinking log_b(x^n) equals (log_b x)^n instead of n·log_b(x)
Did You Know?
This rule is why log scales handle enormous ranges — raising to a power just multiplies the log.
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