Solving Exponential Equations

If both sides can be written with the same base, equate their exponents; the base must be positive and not 1. If the bases do not match conveniently, take logarithms. If you see repeated powers added together, simplify or substitute before taking logs.

Each exponent gives a different power of 2, so equal powers have equal exponents. Check: $4^3=8^2=64$.

Solve $9^{x+1}=27^{x-1}$ without a calculator.

Both sides are positive. Take $\ln$ of both sides, then use the power law on the whole exponent.

Here “s.f.” means significant figures. You can use base-10 logs instead. Do not mix log and ln across the two sides. Enter the complete numerator and denominator in brackets, without rounding their log values first.