# Solving exponential equations with logarithms

> IB Mathematics: Applications and Interpretation HL · Unit 1: Number and Algebra
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u1-solving-exponential-equations-with-logarithms/

This module covers solving exponential equations with unknown exponents using logarithm properties. We cover same-base, different-base equations, real-world modeling applications, and common exam traps to avoid.

**Prerequisites:** [Laws of logarithms](https://www.owlsprep.com/study/ib-math-ai-hl-u1-laws-of-logarithms/); [Exponential function properties](https://www.owlsprep.com/study/ib-math-ai-hl-u1-exponential-functions/)

## Learning objectives

- Convert exponential equations to logarithmic form to solve for unknown exponents
- Solve exponential equations with different bases and linear exponents
- Identify extraneous solutions and invalid cases
- Apply logarithm methods to solve real-world exponential modeling problems

## 1. Core Method: Taking Logarithms of Both Sides

**Logarithm method for exponential equations** — We can take the logarithm of both sides, use the power rule to bring down the exponent, then isolate and solve for $x$. This works because logarithms are one-to-one, so equality is preserved.

*Notation:* $a^x = b$, $a>0, a\neq 1$, $b>0$

*Example:* $2^x = 10 \implies x = \frac{\log 10}{\log 2}$

Any base of logarithm works for this method: common logarithm (base 10) and natural logarithm (base $e$) are most commonly used, as they are available on all standard calculators.

**Worked example:** Solve $5^x = 32$, give your answer to 3 significant figures.

1. Take the common logarithm of both sides:
2. $$\log\left(5^x\right) = \log(32)$$
3. Apply the power law of logarithms $\log(a^b) = b\log a$ to bring down the exponent:
4. $$x \log 5 = \log 32$$
5. Rearrange to isolate $x$:
6. $$x = \frac{\log 32}{\log 5}$$
7. Evaluate with a calculator to get the final answer:
8. $$x \approx 2.15$$

> **tip**
>
> Always confirm your answer by substituting back into the original equation: $5^{2.15} \approx 32$, which confirms your solution is correct.

*Calculator:* allowed

## 2. Equations with Linear Exponents on Both Sides

Most exam questions have linear exponents on both sides of the equation, often with different bases. We use the same core logarithm method, then rearrange the resulting linear equation to collect like terms and solve for the unknown.

**Worked example:** Solve $3^{2x - 1} = 7^{x}$, give your answer to 2 decimal places.

1. Take the natural logarithm of both sides (any base works):
2. $$\ln\left(3^{2x-1}\right) = \ln\left(7^x\right)$$
3. Apply the power law to bring down both exponents:
4. $$(2x - 1)\ln 3 = x \ln 7$$
5. Expand the left-hand side and rearrange to collect $x$ terms on one side:
6. $$2x \ln 3 - \ln 3 = x \ln 7$$
7. $$2x \ln 3 - x \ln 7 = \ln 3$$
8. Factor $x$ out of the left-hand side:
9. $$x\left(2\ln 3 - \ln 7\right) = \ln 3$$
10. Isolate $x$ and evaluate with a calculator:
11. $$x = \frac{\ln 3}{2\ln 3 - \ln 7} \approx 4.79$$

> **warning**
>
> If you get a negative coefficient when factoring, keep the negative sign in the denominator, do not incorrectly cancel it out.

*Calculator:* allowed

## 3. Solving Exponential Equations in Context

IB AI HL regularly asks you to solve for unknown time or growth rate in exponential growth and decay models. These problems follow the same logarithm method, but require you to interpret your answer in the context of the question, including unit conversion where needed.

**Worked example:** A bacteria population grows according to $P(t) = 100e^{0.12t}$, where $P(t)$ is population at time $t$ hours. How long does it take for the population to reach 500? Give your answer to the nearest minute.

1. Set the population equal to 500:
2. $$500 = 100 e^{0.12t}$$
3. Divide both sides by 100 to isolate the exponential term:
4. $$5 = e^{0.12t}$$
5. Take the natural logarithm of both sides (inverse of $e^x$):
6. $$\ln 5 = \ln\left(e^{0.12t}\right)$$
7. Simplify using the inverse property $\ln(e^a) = a$:
8. $$\ln 5 = 0.12 t$$
9. Solve for $t$ in hours:
10. $$t = \frac{\ln 5}{0.12} \approx 13.412 \text{ hours}$$
11. Convert the decimal part of the hour to minutes: $0.412 \times 60 \approx 25$ minutes.
12. Final answer: 13 hours 25 minutes

**Check your understanding**

Test your understanding:

1. A radioactive substance decays as $m(t) = 200(0.95)^t$ where $m$ is mass in grams, $t$ is days. What is the first step to find the half-life?

   - Set $100 = 200(0.95)^t$, then take logs of both sides
   - Set $t = 0.5$ and solve for $m$
   - Divide by 200 immediately before anything else

   *Answer:* Set $100 = 200(0.95)^t$, then take logs of both sides

   *Why:* Correct! Half-life means the mass is half the original starting mass, so first substitute $m(t) = 100$, then simplify and take logs.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Taking the logarithm of a negative number and proceeding to calculate a solution
  - Why it fails: Logarithms of negative numbers are not real, so the equation has no real solution
  - Correct: Always check that both sides of the equation are positive before taking logarithms; if one side is negative, state there are no real solutions.
- **Wrong:** Applying the power rule to a sum of powers: $\log(a^x + b^y) = x\log a + y\log b$
  - Why it fails: The power rule only applies to powers inside a logarithm, not sums of powers
  - Correct: Always isolate the entire exponential term on one side of the equation before taking the logarithm of both sides.
- **Wrong:** Simplifying $\frac{\log a}{\log b} = \log a - \log b = \log\left(\frac{a}{b}\right)$
  - Why it fails: Confuses fraction division with the logarithm quotient rule
  - Correct: Remember $\frac{\log a}{\log b} = \log_b a$, calculate the numerator and denominator separately before dividing.
- **Wrong:** Leaving the final answer in the wrong units for context problems
  - Why it fails: Rushing to write the answer after calculating the decimal value
  - Correct: Always check what units the question asks for, and convert decimal values to the required units before finishing.

## Cheatsheet

| Equation Type | Steps | Final Form |
| --- | --- | --- |
| Single term $a^x = b$ | 1. Take log both sides 2. Bring down x | $x = \frac{\log b}{\log a}$ |
| Linear exponents both sides $a^{mx+c} = b^{nx+d}$ | 1. Take log both sides 2. Expand 3. Collect x terms | $x = \frac{d\log b - c\log a}{m\log a - n\log b}$ |
| Natural base model $A = A_0 e^{kt}$ | 1. Divide by $A_0$ 2. Take $\ln$ both sides | $t = \frac{1}{k}\ln\left(\frac{A}{A_0}\right)$ |
| Right-hand side is negative | Stop immediately | No real solutions |

## What's next

Solving exponential equations with logarithms is a foundational skill for all further work with exponential and logarithmic modeling, a core theme in IB AI HL. You will use this skill constantly when working with compound interest, population growth, radioactive decay, and regression models for exponential data. This technique appears in both Paper 1 and Paper 2 exam questions, so mastering both algebraic manipulation and calculator evaluation is key for full marks. Even when using a GDC to solve these equations, understanding the logarithm method helps you check your answers and catch mistakes.

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