Recipe

Solve an exponential equation

Use when: the unknown sits in an exponent, anywhere in the equation

Goal: every exact value of the unknown, checked against the original equation

Taught in Solving exponential and logarithmic equations

Quick recipe

Isolate the exponential term
Check for a hidden quadratic
Match bases, or take a log
Solve the resulting equation for the unknown
Check every candidate in the original equation

1
Get the exponential expression by itself on one side before touching anything else. Clear any coefficient multiplying it or constant added to it.

Example

$5\cdot2^x=240$ becomes $2^x=48$ after dividing both sides by 5.

Why: a log or a base-matching move only works on an equation already in the pure shape $b^{(\dots)} = (\text{something})$ . A stray factor breaks that shape and forces an extra step later to undo it.
2
Look for $b^{2x}$ (or $(b^x)^2$ ) sitting alongside a plain $b^x$ term in the same equation.

Example

$4^x-3\cdot2^x-4=0$ has $4^x=(2^x)^2$ , so $u=2^x$ turns it into $u^2-3u-4=0$ , giving $u=4$ or $u=-1$ ; the second root is discarded before continuing.
- only one exponential term appears — skip ahead to Step 3.
- $b^{2x}$ and $b^x$ both appear — substitute $u=b^x$ , solve the resulting quadratic in $u$ , and discard any root with $u\le0$ .
Why: an exponential expression can never be zero or negative, so a nonpositive "solution" for $u$ never corresponds to a real value of $x$ . Catching this now avoids chasing a dead end through the rest of the recipe.
3
Check whether both sides can be rewritten as powers of the same base.
- both sides are (or can be rewritten as) powers of the same base — rewrite them that way and set the exponents equal, using the one-to-one property from earlier in this section.
- no common base is available — take a logarithm of both sides (any base; $\ln$ is conventional) and use the power rule to bring the exponent down as a multiplier.
Why: matching bases gives an exact answer with no calculator; logging always works and is the fallback the moment a common base doesn't exist.
4
What's left after either move is an ordinary linear equation in the unknown. Finish it the way any linear equation gets solved.

Why: isolating the exponent by itself, or the log by the power rule, always leaves nothing but ordinary algebra behind.
5
Substitute each surviving value back into the equation as it was written before any substitution or logging.

Why: every individual move above is reversible for an honest exponential equation, so this step is mainly an arithmetic safety net. When Step 2 produced more than one branch, it's also how you confirm each branch's answer actually works.

Common mistakes

Taking a log before isolating the exponential term: logging $5\cdot2^x$ directly needs an extra product-rule step that logging $2^x$ alone doesn't.
Forgetting to discard a nonpositive root from a quadratic-in-form substitution.
Reading $\ln(3^x)$ as $(\ln3)^x$ instead of $x\ln3$ ; see the common-mistake box above.