Product Rule for Logarithms
The product rule for logarithms states that the logarithm of a product equals the sum of the logarithms of the factors.
Formula
\log_b(xy) = \log_b(x) + \log_b(y)
Definition
The product rule says the log of a product equals the sum of the logs: for any valid base $b$ and positive $x, y$, $\log_b(xy) = \log_b(x) + \log_b(y)$. This follows because $b^m \cdot b^n = b^{m+n}$: if $\log_b(x) = m$ and $\log_b(y) = n$, then $xy = b^m \cdot b^n = b^{m+n}$, so $\log_b(xy) = m + n$. More deeply, the product rule is the defining property of logarithms as group homomorphisms from $(\mathbb{R}^+, \times)$ to $(\mathbb{R}, +)$, and any continuous (or even just measurable) function satisfying $f(xy) = f(x) + f(y)$ must be a logarithm, a consequence of Cauchy's functional equation; without a regularity condition, pathological solutions exist via Hamel bases of $\mathbb{R}$ over $\mathbb{Q}$.
Example
$\log_2(4 \times 8) = \log_2(4) + \log_2(8) = 2 + 3 = 5$, matching $\log_2(32) = 5$ since $2^5 = 32$. To solve $\ln(x) + \ln(x - 3) = \ln(10)$: combine to $\ln(x(x-3)) = \ln(10)$, so $x(x-3) = 10$, giving $x^2 - 3x - 10 = 0$, $(x-5)(x+2) = 0$, and since $x > 3$ is required by the domain, $x = 5$. The product rule yields $\log(n!) = \sum_{k=1}^{n} \log(k)$, enabling Stirling's approximation via the integral of log, connecting the discrete product (factorial) to continuous analysis (integration).
Key Insight
This rule is what made logarithm tables useful for centuries, to multiply two large numbers, just look up their logs, add, and look up the antilog of the sum, turning multiplicative equations into additive ones that are easier to solve; this is exactly how slide rules work, physical addition of log-scale lengths computes multiplication.