Exponential and Logarithmic Functions
Core concepts
Exponential functions y = a·bˣ have constant multiplicative growth; logarithms are their inverses. You must know the laws of logarithms (product, quotient, power, change of base), the graphs of y = eˣ and y = ln x, and how to solve equations by taking logs of both sides or exponentiating.
Solving exponential equations often means rewriting both sides with a common base, or applying a logarithm to isolate the exponent. Solving logarithmic equations means combining logs and converting to exponential form, but always checking that arguments remain positive. Growth and decay models, half-life, and compound interest all reduce to the same exponential structure.
Traps to watch
- Writing log(a + b) = log a + log b, which is false; the product law applies only to multiplication.
- Forgetting that the domain of ln x is x > 0, so some algebraic solutions must be rejected.
- Mishandling the coefficient when rewriting, e.g. 2 log x is log x², not (log x)².
- Confusing a horizontal asymptote of an exponential graph with a vertical one of a log graph.
Rule of thumb
To undo an exponent, take a logarithm; to undo a logarithm, exponentiate. After solving, substitute back to check the original domain.
Worked example
Solve 3^(x+1) = 20. Take logs: (x + 1) log 3 = log 20, so x + 1 = log 20 / log 3 ≈ 2.727, giving x ≈ 1.727. Check by substitution: 3^2.727 ≈ 20, confirming the answer.