# Solving exponential equations using logarithms

> IB Mathematics Applications and Interpretation SL · Number and Algebra
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u1-solving-exponential-equations-using-logarithms/

This sub-topic teaches you to solve equations where the unknown variable appears in the exponent, using logarithm properties. You will learn methods for different bases and apply these techniques to common exam problems.

**Prerequisites:** [Properties of logarithms](https://www.owlsprep.com/study/ib-math-ai-sl-u1-logarithm-properties/); [Laws of exponents](https://www.owlsprep.com/study/ib-math-ai-sl-u1-exponent-laws/)

## Learning objectives

- Isolate exponential terms in preparation for solving
- Apply logarithms to solve for unknown exponents
- Solve exponential equations with base 10 and base e
- Interpret solutions for exponential equations in real-world context

## Isolating the exponential term

Before you can apply logarithms to solve an exponential equation, you must first rearrange the equation to isolate the term with the unknown exponent on one side of the equals sign. All constant terms must be moved to the opposite side and simplified.

**Isolated exponential term** — An exponential term that is alone on one side of an equation, with no added constants and a leading coefficient of 1. For $a \times b^{kx + c} + d = e$, the isolated form is $b^{kx + c} = \frac{e-d}{a}$.

*Example:* $2^{x+1} = 8$ is isolated, $2^{x+1} + 5 = 13$ is not.

**Worked example:** Isolate the exponential term in the equation $3 \times 2^{x} - 4 = 20$

1. Add 4 to both sides to move constants to the right-hand side:
2. $$3 \times 2^x = 20 + 4$$
3. Simplify the right-hand side:
4. $$3 \times 2^x = 24$$
5. Divide both sides by 3 to isolate the exponential term:
6. $$2^x = 8$$

## Taking logarithms of both sides

Once the exponential term is isolated, take the logarithm of both sides to use the logarithm power rule, which brings the unknown exponent down to become a linear multiplier. This converts the exponential equation into a linear equation that can be solved easily.

**Logarithm Power Rule** — For any positive $a$ and real $b$, the logarithm of a term raised to a power equals the power multiplied by the logarithm of the base.

*Notation:* \log(a^b) = b\log(a)

> **tip**
>
> Any base of logarithm works, as long as you use the same base on both sides. Use natural log $\ln$ for equations with base $e$ to simplify calculations.

**Worked example:** Solve $e^{3x} = 12$ to 3 significant figures

1. Since the base is $e$, take the natural logarithm of both sides:
2. $$\ln(e^{3x}) = \ln(12)$$
3. Use the identity $\ln(e^a) = a$ to simplify the left-hand side:
4. $$3x = \ln(12)$$
5. Divide both sides by 3 to isolate $x$:
6. $$x = \frac{\ln(12)}{3} \approx 0.828$$

## Solving contextual exponential equations

IB AI SL exams frequently ask you to solve exponential equations derived from real-world contexts like population growth, compound interest, and radioactive decay. The solving process is identical to abstract equations, but you must interpret your final solution in context.

**Worked example:** A population of bacteria grows by $P(t) = 500e^{0.1t}$, where $P(t)$ is the number of bacteria after $t$ minutes. Find when the population reaches 2000, to 1 decimal place.

1. Substitute $P(t) = 2000$ into the model:
2. $$2000 = 500e^{0.1t}$$
3. Divide both sides by 500 to isolate the exponential term:
4. $$4 = e^{0.1t}$$
5. Take the natural logarithm of both sides and simplify:
6. $$\ln(4) = 0.1t$$
7. Solve for $t$:
8. $$t = 10\ln(4) \approx 13.9$$
9. Interpret the solution: The population reaches 2000 after ~13.9 minutes.

## Common pitfalls

- **Wrong:** Taking logarithms before isolating the exponential term
  - Why it fails: $\log(2^x + 3) \neq \log(2^x) + \log(3)$, so the power rule cannot be applied correctly
  - Correct: Always rearrange to isolate the exponential term before taking logarithms of both sides
- **Wrong:** Ignoring a negative result after isolating the exponential term
  - Why it fails: Logarithms are only defined for positive inputs, so a negative right-hand side means no solution
  - Correct: Check the sign after isolating; if negative, state there are no real solutions
- **Wrong:** Mixing logarithm bases when calculating
  - Why it fails: Inconsistent bases lead to incorrect values for the unknown variable
  - Correct: Use the same logarithm base for both sides of the equation
- **Wrong:** Rounding intermediate steps too early
  - Why it fails: Premature rounding leads to inaccuracies in the final answer that lose exam marks
  - Correct: Keep full calculator precision until the final step, then round to the required significant figures

## Cheatsheet

| Step | Action | Example |
| --- | --- | --- |
| 1 | Isolate exponential term | $3 \times 2^x - 5 = 19 \to 2^x = 8$ |
| 2 | Take log of both sides | $\log(2^x) = \log(8)$ |
| 3 | Apply power rule | $x\log(2) = \log(8)$ |
| 4 | Solve for $x$ | $x = \frac{\log(8)}{\log(2)}$ |
| For base $e$ | Use $\ln$, $\ln(e^{kx}) = kx$ | $e^{2x} = 10 \to 2x = \ln(10) \to x \approx 1.15$ |

## What's next

Now that you can solve exponential equations using logarithms, you can apply this core skill to a wide range of real-world modeling problems common in IB AI SL. Exponential models feature throughout the course, used for topics including population growth, compound interest, asset depreciation, and radioactive decay, so this solving technique is foundational for many Paper 1 and Paper 2 questions. This skill also connects to working with logarithms in other contexts, such as linearizing exponential models for regression analysis. You can extend your knowledge and practice with the follow-up topics below.

- [Approximation, absolute and percentage error](https://www.owlsprep.com/study/ib-math-ai-sl-u1-approximation-absolute-and-percentage-error/)
- [Functions](https://www.owlsprep.com/study/ib-math-ai-sl-u2-overview/)

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