Properties of logarithms

Every logarithm rule is an exponent rule in disguise. The definition log_b(x) = y is equivalent to b^y = x — a logarithm answers the question "what exponent turns the base into this number?" Because logarithms are exponents, the familiar exponent laws translate directly, which is why the log of a product becomes a sum and the log of a power becomes a coefficient.

Four rules do nearly all the work. The product rule: logb(xy) = logb(x) + logb(y). The quotient rule: logb(x/y) = logb(x) − logb(y). The power rule: logb(xⁿ) = n · logb(x). And the change of base formula: logb(x) = log(x) / log(b), which lets you evaluate a logarithm of any base on a calculator that only offers base 10 and base e. Three identities are worth memorizing alongside them: logb(1) = 0, logb(b) = 1, and logb(bˣ) = x.

Sign errors and invented rules cause most mistakes. There is no rule for the log of a sum — log(x + y) does not simplify, and it is definitely not log(x) + log(y). Likewise log(x)/log(y) is not log(x) − log(y); the quotient rule applies to the log of a quotient, not to a quotient of logs. Watch the power rule's scope as well: log(x²) equals 2·log(x), but (log x)² does not. Finally, the argument of a logarithm must be positive, so check the solutions of any logarithmic equation back in the original and discard those that make an argument zero or negative.

In practice these rules are how you solve equations with the variable in an exponent: take the log of both sides, use the power rule to bring the exponent down as a coefficient, and solve the resulting linear equation. The ACT math section tests logarithm properties in its intermediate algebra content, usually by asking you to condense an expression into a single logarithm, expand one into separate terms, evaluate a log without a calculator, or convert between logarithmic and exponential form.