Solve equations of the form a^x = b using logarithms or other appropriate methods

What are Exponential Equations?

An exponential equation has the form $a^x = b$, where $a$ (the base) and $b$ are known numbers, and $x$ is the unknown. Think of it like a plant growing: the base $a$ is the growth factor, and the exponent $x$ tells you how many times the plant has doubled (or tripled, etc.).

Using Logarithms to Solve $a^x = b$

The key trick is to take the logarithm of both sides. Any logarithm works, but we usually use the natural log $\ln$ or the common log $\log_{10}$ because calculators have them built‑in. The steps are:

1. Write the equation: $a^x = b$.
2. Take the logarithm of both sides: $\log(a^x) = \log(b)$.
3. Use the power rule for logs: $x\,\log(a) = \log(b)$.
4. Isolate $x$: $x = \dfrac{\log(b)}{\log(a)}$.

🎓 Tip: If you’re using a calculator, just type log(b)/log(a) or ln(b)/ln(a) to get the answer instantly.

Example 1: Solve $2^x = 32$

We know that $32 = 2^5$, so we can guess that $x = 5$. Let’s confirm with logs:

$$x = \dfrac{\log(32)}{\log(2)} = \dfrac{1.50515}{0.30103} \approx 5$$

?? The answer is $x = 5$.

Example 2: Solve $5^x = 125$

Recognise that $125 = 5^3$, so $x = 3$. Using logs:

$$x = \dfrac{\log(125)}{\log(5)} = \dfrac{2.09691}{0.69897} \approx 3$$

?? The answer is $x = 3$.

Example 3: Solve $3^x = 81$

Since $81 = 3^4$, we expect $x = 4$. Check with logs:

$$x = \dfrac{\log(81)}{\log(3)} = \dfrac{1.90849}{0.47712} \approx 4$$

?? The answer is $x = 4$.

📚 Examination Tips

• Always check if the numbers are powers of each other before using a calculator.
• Remember the power rule: $\log(a^x) = x\,\log(a)$.
• When the base is $e$, use natural logs ($\ln$) for a cleaner calculation.
• For multiple-choice questions, try to estimate the answer using known powers.
• Show all steps clearly; examiners look for the process, not just the final answer.