AHL 1.9 (HL)—Logarithm laws

For positive arguments and a valid base b, logarithms obey log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x − log_b y, and log_b(xᵏ) = k log_b x. These rules convert multiplication, division and powers into simpler additive relationships.

To solve 5ˣ = 17, take natural logs: x ln 5 = ln 17, so x = ln 17/ln 5. The same change-of-base idea works with a calculator that provides ln or log.

The product rule does not apply to a sum: log(a + b) is not log a + log b. Every logarithm argument must be positive, and the base must be positive and not equal to 1.

IB examination boundary: the logarithm base is $10$ or $e$, so use $\log$ or $\ln$ consistently. Example: $\ln(3x)-\ln x=\ln3$ for $x>0$; the domain condition cannot be discarded even if the simplified expression is constant. The laws apply only when every original logarithm argument is positive.