Power Rule for Logarithms
Functions & Advanced AlgebraThe power rule for logarithms states that the logarithm of a number raised to an exponent equals the exponent times the logarithm of the number.
Formula
\log_b(x^r) = r \log_b(x)
Definition
The power rule says: log of something raised to a power = that power times the log. The exponent comes down in front of the log.
Example
$\log_2(8^3) = 3 \log_2(8) = 3 \times 3 = 9$. Check: $8^3 = 512$ and $\log_2(512) = 9$ (since $2^9 = 512$). Correct.
Key Insight
Moving the exponent in front is the most powerful logarithm rule for solving equations. Whenever a variable is in an exponent, take a log and bring the exponent down where you can work with it.
Definition
For valid base $b$, positive $x$, and any real $r$: $\log_b(x^r) = r \log_b(x)$. Proof: if $y = \log_b(x)$, then $x = b^y$, so $x^r = b^{ry}$, thus $\log_b(x^r) = ry = r \log_b(x)$.
Example
Solve $5^{2x} = 200$: take log of both sides: $\log(5^{2x}) = \log(200)$. $2x \log(5) = \log(200)$. $2x = \log(200)/\log(5) \approx 2.301/0.699 \approx 3.292$. $x \approx 1.646$.
Key Insight
The power rule is what lets us "bring down" exponents when solving exponential equations. Without it, $x = 5$ in $3^x = 5$ would be impossible to isolate. With it: $x = \log(5)/\log(3)$.
Definition
The power rule $\log(x^r) = r\log(x)$ holds for real $r$ and $x > 0$. It follows from the homomorphism property applied to repeated multiplication: $\log(x^n) = \log(x \cdot x \cdots x) = n\log(x)$, extended to real $r$ by continuity. For complex logarithms, $\log(z^r) = r\log(z)$ requires care about branch cuts.
Example
Complex: $\ln((-1)^2) = \ln(1) = 0$, but $2\ln(-1) = 2i\pi$ (principal branch). These differ, showing the power rule fails naively for complex numbers with certain exponents. Branch cuts must be handled carefully.
Key Insight
The failure of the power rule for complex numbers reflects the multivaluedness of the complex logarithm. Correct statements require specifying a branch. This subtlety has caused errors in formal proofs and is a key example of why complex analysis requires more care than real analysis.