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Exponential equations have the variable in the exponent. Two main strategies: (1) If both sides can be written with the same base, set exponents equal (e.g., 2x = 8 becomes 2x = 23, so x = 3). (2) If not, take the logarithm of both sides and use log properties to isolate the variable (e.g., 3x = 20 becomes x = log(20)/log(3)). Strategy 2 is the general method and works for all cases.
Start with equations solvable by rewriting with a common base. Then introduce the "take log of both sides" technique for equations that cannot be simplified to a common base. Practice with various bases including e. Apply to real-world problems (population doubling time, radioactive decay half-life).
Solving an exponential equation means finding the value of a variable that appears in an exponent. You already know from your work with exponential functions that the graph of y = bˣ is one-to-one — it passes the horizontal line test — so each output corresponds to exactly one input. That one-to-one property is what guarantees exponential equations have unique solutions and is what both solution strategies exploit.
The common base strategy works when both sides of the equation can be written as powers of the same base. For example, 4ˣ = 32 becomes 2²ˣ = 2⁵, so 2x = 5 and x = 5/2. This works because if bᵐ = bⁿ and b ≠ 1, then m = n — identical outputs from the same one-to-one function mean identical inputs. The skill is rewriting numbers as powers: 4 = 2², 8 = 2³, 27 = 3³, 1/9 = 3⁻², and so on. When you can do this, you reduce the exponential equation to a linear or polynomial equation.
The logarithm strategy handles the cases where a common base isn't obvious — which is most real-world cases. The equation 3ˣ = 20 has no convenient common base, so you apply log to both sides: log(3ˣ) = log(20). Using the power rule for logarithms (which you know: logₐ(mⁿ) = n·logₐ(m)), this becomes x·log(3) = log(20), so x = log(20)/log(3) ≈ 2.727. The power rule is what allows you to bring the exponent down as a coefficient — but only after taking the logarithm, never before. The logarithm "undoes" the exponential the same way division undoes multiplication: they are inverse operations.
A practical guide: for equations with base e, use the natural logarithm (ln) because ln(eˣ) = x exactly. For all other bases, any logarithm works (the ratio log(b)/log(a) is the same regardless of which log base you use), but base-10 log or ln are the most convenient. The biggest trap is misapplying log properties to sums: log(2ˣ + 5) is not x·log(2) + log(5). Log properties only simplify products, quotients, and powers — not sums. If you see a sum inside a logarithm, or a sum of exponential terms, the expression usually requires a substitution or a different approach entirely.