Solving exponential equations with logarithms

IB Mathematics: Applications and Interpretation HL· 1.6 Exponential and logarithmic equations· 35 min read

1. 1. Core Method: Taking Logarithms of Both Sides★★☆☆☆⏱ 15 min

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📘 Definition

Logarithm method for exponential equations

$a^{x} = b$ , $a > 0, a \neq = 1$ , $b > 0$

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.

Example:

$2^{x} = 10 ⟹ x = \frac{l o g 10}{l o g 2}$

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 $lo g (a^{b}) = b lo g a$ 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
$x \approx 2.15$

2. 2. Equations with Linear Exponents on Both Sides★★★☆☆⏱ 20 min

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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 $3^{2 x - 1} = 7^{x}$ , 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
$x = \frac{\ln 3}{2\ln 3 - \ln 7} \approx 4.79$

3. 3. Solving Exponential Equations in Context★★★☆☆⏱ 20 min

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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 $P (t) = 100 e^{0.12 t}$ , 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

✓ Quick check

Test your understanding:

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

4. Common Pitfalls

Wrong move:

Taking the logarithm of a negative number and proceeding to calculate a solution

Why:

Logarithms of negative numbers are not real, so the equation has no real solution

Correct move:

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 move:

Applying the power rule to a sum of powers: $lo g (a^{x} + b^{y}) = x lo g a + y lo g b$

Why:

The power rule only applies to powers inside a logarithm, not sums of powers

Correct move:

Always isolate the entire exponential term on one side of the equation before taking the logarithm of both sides.

Wrong move:

Simplifying $\frac{l o g a}{l o g b} = lo g a - lo g b = lo g (\frac{a}{b})$

Why:

Confuses fraction division with the logarithm quotient rule

Correct move:

Remember $\frac{l o g a}{l o g b} = lo g_{b} a$ , calculate the numerator and denominator separately before dividing.

Wrong move:

Leaving the final answer in the wrong units for context problems

Why:

Rushing to write the answer after calculating the decimal value

Correct move:

Always check what units the question asks for, and convert decimal values to the required units before finishing.

5. Quick Reference Cheatsheet

Equation Type	Steps	Final Form
Single term $a^{x} = b$	Take log both sides 2. Bring down x	$x = \frac{l o g b}{l o g a}$
Linear exponents both sides $a^{m x + c} = b^{n x + d}$	Take log both sides 2. Expand 3. Collect x terms	$x = \frac{d l o g b - c l o g a}{m l o g a - n l o g b}$
Natural base model $A = A_{0} e^{k t}$	Divide by $A_{0}$ 2. Take $ln$ both sides	$t = \frac{1}{k} ln (\frac{A}{A _{0}})$
Right-hand side is negative	Stop immediately	No real solutions

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Solve for x in $2^{x} = 15$
2023 · 2
Exponential decay find time to halve
2022 · 1
Solve $3^{2 x - 1} = 7^{x}$

Going deeper

formula sheetIB AI HL Formula Booklet ExcerptLogarithm laws are given in the formula booklet for exams

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.