Exponents and Logarithms: Laws, Equations, and Models
Exponential and Logarithmic Reasoning
An exponential function y = a·bˣ has a constant ratio; b > 1 gives growth, 0 < b < 1 gives decay, and the multiplier is 1 + r for a growth rate r. The base e appears in continuous growth models. The defining equivalence is y = log_b x exactly when bʸ = x, with b > 0, b ≠ 1, and x > 0.
The laws of logarithms mirror exponent laws: log(MN) = log M + log N, log(M/N) = log M − log N, and log(Mᵖ) = p·log M. Change of base, log_b M = log M / log b, lets you evaluate anything on a calculator. Solve exponential equations by rewriting both sides with a common base when possible, otherwise take the logarithm of both sides. Solve logarithmic equations by condensing to a single logarithm and converting to exponential form.
Always verify candidate solutions in the original equation. Because a logarithm only accepts positive arguments and bˣ is always positive, some algebraic solutions are extraneous.
Traps: writing log(M + N) = log M + log N (false); applying the power law to log(M + N); forgetting that log x² = 2 log x requires x > 0; missing extraneous roots; confusing the base of the exponent with a leading coefficient.
Worked example: Solve log₂x + log₂(x − 2) = 3. Condense: log₂[x(x − 2)] = 3, so x² − 2x = 8, giving x = 4 or x = −2. Since x must exceed 2, only x = 4 survives.
Rule of thumb: exponent laws inside, log laws outside - and check every answer in the original equation.