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Overview
Pre-Calculus 12
Relations and Functions
14. Demonstrate understanding of product, quotient, and power laws of logarithms
14.4 Power rule of logarithms
Power Rule of Logarithms
See how to rewrite the log of a power as a coefficient times a log, with proof, examples, and common mistakes to avoid.
What You'll Learn
The power rule of logarithms states that the log of a number raised to a power equals the exponent times the log of the number.
In symbols, \(\log_b(x^n) = n \log_b(x)\), and it works for any valid base and any real exponent \(n\).
The rule comes directly from the power of a power rule for exponents, so the two ideas mirror each other.
Expanding means pulling the exponent out front; condensing means running the same steps in reverse.
The rule only applies when the base of the power and the number inside the log match the same quantity being raised.
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