Exponentials and Logarithms
Core concepts
Know the laws of logarithms, the relationship between exponential and logarithmic functions, the graphs of y = eˣ and y = ln x, and the use of logarithms to solve equations of the form aˣ = b and to linearise power and exponential laws for graphical analysis.
Traps
- log(A + B) is not log A + log B. The product law applies to multiplication only.
- When taking logs of both sides, apply the log to the whole side, not term by term.
- The domain of ln x is x > 0; reject any solution that makes an argument non-positive.
- For y = axⁿ, plotting log y against log x gives gradient n and intercept log a; for y = abˣ, plotting log y against x gives gradient log b and intercept log a. Mixing these up is common.
Worked example
Solve 3^(2x − 1) = 5. Take natural logs: (2x − 1) ln 3 = ln 5, so 2x − 1 = ln 5 / ln 3, giving x = ½(1 + ln 5 / ln 3) ≈ 1.232. Rule of thumb: when the unknown is in the exponent, take logs of both sides and bring the exponent down using the power law.