Power Rule of Logarithms
Simple Explanation
The logarithm of a number raised to a power equals the power times the logarithm of the number: log_b(Mᵖ) = p · log_b M. This lets you bring an exponent down out of a logarithm as an ordinary multiplier.
Why Do We Need It?
The power rule is the key tool for solving equations where the unknown sits in an exponent — it converts an exponent into a coefficient, which ordinary algebra can then isolate.
Formula
Power Rule of Logarithms
log_b(Mᵖ) = p · log_b M
The logarithm of a number raised to a power equals the power times the logarithm of the number.
- b
- — the base of the logarithm (b > 0, b ≠ 1)
- M
- — a positive real number
- p
- — the exponent applied to M
When to use it: Whenever a logarithm's argument is itself a power, letting you bring the exponent down as a multiplier — essential for solving equations where the variable is in an exponent.
Worked Example
Use the power rule to simplify
Simplify log₃(9²) using the power rule, then evaluate.
Why Does This Work?
If log_b M = q, then M = bᵠ, so Mᵖ = (bᵠ)ᵖ = bᵖᵠ by the power-of-a-power exponent rule — meaning log_b(Mᵖ) is exactly pq, i.e. p times the original logarithm.
Real-Life Example
Solving for time in a growth formula
An investment formula A = A₀(1.06)ᵗ needs to be solved for the exponent t.
Taking a logarithm of both sides brings t down using the power rule, turning an "unknown exponent" problem into ordinary algebra.
Practice
Evaluate log₂(4³) using the power rule (give the final numeric value).
MediumCommon mistake
Applying the power rule to the base instead of the argument — log_b(Mᵖ) = p·log_b M works when the exponent is on M (the argument); it says nothing about changing the base b itself.
Quick Review
- log_b(Mᵖ) = p · log_b M.
- The power rule brings an exponent down out of a logarithm as a multiplier.
- This is the key step for solving equations with the variable in an exponent.