Solving Exponential Equations with Logarithms

Solving an exponential equation with unlike bases means taking a logarithm of both sides, pulling the exponent down with the power rule, and solving what remains.

Solve 3ˣ = 10. No shared base rescues this one, so take the common log of both sides: log(3ˣ) = log(10). The power rule pulls the unknown down: x·log(3) = 1. Divide: x = 1/log(3) ≈ 2.096. The check runs forward: 3^2.096 ≈ 10, agreement to the rounding. The same three moves solve every stubborn exponential — log both sides, pull down, divide — and the divide step lands on a quotient of logs, which is change-of-base arithmetic wearing work clothes, not the log of any quotient.