Solving any exponential equation in 3 steps

Published 2026-04-15 · Written by Pete Bromfield

The same 3-step recipe solves every exponential-equation question you'll meet at A-Level. Here it is.

The 3 steps

1. Isolate the exponential on one side.
2. Take logs of both sides (any base — usually $\ln$ or $\log_{10}$).
3. Use $\log(a^b) = b \log a$ to bring the exponent down, then solve linearly for $x$.

Worked example

Solve $2 \cdot 3^{x} = 15$.

1. $3^{x} = 7.5$.
2. $\ln(3^x) = \ln 7.5$.
3. $x \ln 3 = \ln 7.5$, so $x = \dfrac{\ln 7.5}{\ln 3} \approx 1.834$.

Common trap

Students write $\log(2 \cdot 3^x) = \log 2 + \log 3^x = \log 2 + x \log 3$ and end up with a longer, uglier arithmetic path. Isolate the exponential first, then take logs. Every time.

Drill this on the Flashcards page → the "log-law flash" deck has 25 cards on this exact pattern.