Product and Power Properties of Logs

A logarithm answers the question: what exponent do I need? When you multiply two numbers, you're really adding their exponents on a common base, so it makes sense that taking a log of a product gives you a sum. That's the whole idea behind log_b(xy) = log_b(x) + log_b(y): multiplication upstairs becomes addition downstairs, because logs exist to translate between those two worlds.

The mechanism is just exponent rules in disguise. If x = b^m and y = bⁿ, then xy = b^m+n, and taking log base b of both sides gives log_b(xy) = m + n = log_b(x) + log_b(y). The power property works the same way: xⁿ = (b^m)ⁿ = b^mn, so log_b(xⁿ) = n times log_b(x) — the exponent drops down and becomes a coefficient out front. Division follows the same logic in reverse, turning into subtraction, since dividing means subtracting exponents.

These two properties are the tools you use to rewrite a single complicated log as a sum of simpler pieces, or to collapse a sum of logs back into one expression — both directions matter. They also explain graph behavior: stretching the input of a log function horizontally just adds a constant log_b(k) to every output, which is why a horizontal dilation of a log graph looks identical to a vertical shift. That link between algebraic rewriting and graph transformations is where this topic is headed next.