Solving Exponential Equations
Look for a common base first
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.
Worked example
Solve 4ˣ = 8ˣ⁻¹.
Each exponent gives a different power of 2, so equal powers have equal exponents. Check: .
Solve without a calculator.
Show worked answer
Logarithms turn an exponent into a coefficient
Worked example
Solve 5²ˣ⁺¹ = 7.
Both sides are positive. Take 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.
Worked example
Solve 6²ˣ⁻¹ = 5e³ˣ⁺².
(9709/22/M/J/24 Q2) asks for logarithmic working and 4 significant figures. The right side is a product, so its logarithm is a sum.
Check the unrounded value in the original equation, or compare the logarithms of both sides when the powers are large.
Solve , giving an exact expression and 3 significant figures.