Exponential Equation

An exponential equation is an equation where the unknown variable appears in the exponent. To solve one, you typically use logarithms to bring the variable down from the exponent so you can isolate it.

An exponential equation is an equation in which one or more terms contain a variable expression as an exponent. These equations take forms such as $a^{f(x)} = b$ or $a^{f(x)} = c^{g(x)}$ . When both sides can be rewritten with the same base, the exponents can be set equal directly. Otherwise, logarithms are applied to both sides to solve for the variable.

Key Formula

a^{x} = b \implies x = \frac{\log b}{\log a}

Where:

$a$ = the base of the exponential expression (positive, not equal to 1)
$x$ = the unknown variable in the exponent
$b$ = the value the exponential expression equals (must be positive)

Worked Example

Problem: Solve for x:

3^{2x-1} = 54

Step 1: Take the logarithm of both sides. You can use any base — common log (base 10) works fine.

\log(3^{2x-1}) = \log(54)

Step 2: Use the power rule of logarithms to bring the exponent down in front.

(2x - 1)\log 3 = \log 54

Step 3: Divide both sides by

\log 3

to isolate the expression containing

x

2x - 1 = \frac{\log 54}{\log 3} \approx \frac{1.7324}{0.4771} \approx 3.6309

Step 4: Solve the resulting linear equation for

x

2x = 4.6309 \implies x \approx 2.3155

Answer:

x \approx 2.316

(rounded to three decimal places).

Why It Matters

Exponential equations model situations where a quantity grows or decays by a constant percentage over time — things like compound interest, population growth, radioactive decay, and the spread of diseases. Being able to solve them lets you answer questions like "How long until my investment doubles?" or "When will a sample of radioactive material decay to a safe level?"

Common Mistakes

Mistake: Trying to solve by dividing the bases instead of using logarithms.

Correction: When the variable is in the exponent, ordinary algebraic operations on the base won't help. You need logarithms (or matching bases) to extract the variable from the exponent.

Mistake: Forgetting to distribute the logarithm result to the entire exponent expression.

Correction: If the exponent is something like

2x - 1

, the power rule gives

(2x - 1)\log a

, not just

2x \cdot \log a

. Keep the full expression together, then solve the resulting equation.

Related Terms

Exponential Function — the function type that appears in these equations
Logarithm — the main tool used to solve exponential equations
Exponent Rules — rules for simplifying exponential expressions before solving