6.3 Solving Exponential Equations

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What is an Exponential Equation?

An exponential equation is an equation where the unknown (the letter you are solving for, usually $x$) appears as a power (exponent).

For example: $2^x = 8 \qquad 3^x = 20 \qquad 5^x = 100$

In the first example, $2^x = 8$, you might already know the answer is $x = 3$ because $2^3 = 8$. But what about $3^x = 20$? You cannot easily spot the answer — this is where logarithms come in.

The Key Tool: Logarithms

A logarithm is the inverse (opposite) of a power. In plain English:

This means: "What power do I raise $a$ to, in order to get $b$?"

However, in exam questions, you will almost always use common logarithms (written as $\log$, base 10) or natural logarithms (written as $\ln$, base $e$). Both work — just be consistent.

The most important log rule you need here is:

This rule lets you bring the power down in front of the log, turning a difficult exponential equation into a straightforward linear equation.

The Method: Solving $a^x = b$

Follow these steps every time:

Step 1: Take $\log$ (or $\ln$) of both sides of the equation.

Step 2: Use the Power Rule to bring $x$ down in front of the log.

Step 3: Divide both sides by the log term to isolate $x$.

Step 4: Use a calculator to find the decimal answer (unless an exact answer is asked for).

Worked Example 1 — Simple Case

Solve $3^x = 20$.

Step 1: Take $\log$ of both sides. $\log(3^x) = \log 20$

Step 2: Apply the Power Rule — bring the power $x$ to the front. $x \log 3 = \log 20$

Step 3: Divide both sides by $\log 3$. $x = \frac{\log 20}{\log 3}$

Step 4: Use a calculator.

✅ Answer: $x \approx 2.73$

Worked Example 2 — Coefficient in Front of $x$

Solve $5^{2x} = 80$.

Step 1: Take $\log$ of both sides. $\log(5^{2x}) = \log 80$

Step 2: Apply the Power Rule. $2x \log 5 = \log 80$

Step 3: Divide both sides by $\log 5$. $2x = \frac{\log 80}{\log 5}$

Step 4: Divide by 2. $x = \frac{\log 80}{2 \log 5}$

Step 5: Use a calculator. $x = \frac{1.9031...}{2 \times 0.6990...} = \frac{1.9031...}{1.3979...} \approx 1.361$

✅ Answer: $x \approx 1.36$ (to 3 s.f.)