Power Rule for Logarithms
The 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 the log of something raised to a power equals that power times the log: 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 $\log_b(x^r) = ry = r \log_b(x)$; 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, and the rule can fail naively for complex numbers with certain exponents.
Example
$\log_2(8^3) = 3 \log_2(8) = 3 \times 3 = 9$, matching $\log_2(512) = 9$ since $2^9 = 512$. To solve $5^{2x} = 200$: take log of both sides, $2x \log(5) = \log(200)$, so $2x \approx 3.292$ and $x \approx 1.646$. In the complex numbers, $\ln((-1)^2) = \ln(1) = 0$, but $2\ln(-1) = 2i\pi$ (principal branch), these differ, showing the power rule fails naively for complex numbers with certain exponents, and branch cuts must be handled carefully.
Key Insight
The power rule is what lets us "bring down" exponents when solving exponential equations, without it, isolating $x$ in $3^x = 5$ would be impossible, but with it, $x = \log(5)/\log(3)$. The failure of the power rule for complex numbers reflects the multivaluedness of the complex logarithm: correct statements require specifying a branch, a subtlety that has caused errors in formal proofs and is a key example of why complex analysis requires more care than real analysis.