Logarithm Power Rule
The log of a power brings the exponent to the front.
Formula
Variables
Example
log(1000)=3log(10)=3
How it works
This rule says that a power inside a logarithm can be rewritten as a multiplier in front of the log. It works because logarithms measure exponents: if x = b^k, then x^n = (b^k)^n = b^(kn), so the exponent n simply multiplies the original logarithm value k. The base b must be positive and not equal to 1, and the argument x must be positive.
When to use it
Use this when simplifying logarithmic expressions, especially before combining or solving log equations. It is also useful in algebra, calculus, and scientific formulas where powers appear inside logs.
Worked example, step by step
Simplify log_2(8^3) log_2(8^3) = 3 log_2(8) Since 2^3 = 8, log_2(8) = 3 So 3 log_2(8) = 3 × 3 = 9 Final answer: log_2(8^3) = 9
Common mistakes to avoid
- multiplying the exponent by the base instead of moving it in front of the log
- using the rule when the logarithm argument is negative or zero
- forgetting that the base of the log stays the same after rewriting
Did You Know?
This rule is why logarithmic scales compress enormous ranges — earthquakes and sound levels both rely on it.
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