Solving exponential equations using logarithms

IB Mathematics Applications and Interpretation SL· Unit 1, Topic 1.7: Exponential and logarithmic equations· 20 min read

1. Isolating the exponential term★★☆☆☆⏱ 5 min

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.

📘 Definition

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 $a \times b^{k x + c} + d = e$ , the isolated form is $b^{k x + c} = \frac{e - d}{a}$ .

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
$2^x = 8$

2. Taking logarithms of both sides★★★☆☆⏱ 7 min

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.

📘 Definition

Logarithm Power Rule

\log(a^b) = b\log(a)

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.

📐 Worked Example

Solve $e^{3 x} = 12$ 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
$x = \frac{\ln(12)}{3} \approx 0.828$

3. Solving contextual exponential equations★★★☆☆⏱ 6 min

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 $P (t) = 500 e^{0.1 t}$ , 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.

4. Common Pitfalls

Wrong move:

Taking logarithms before isolating the exponential term

Why:

$lo g (2^{x} + 3) \neq = lo g (2^{x}) + lo g (3)$ , so the power rule cannot be applied correctly

Correct move:

Always rearrange to isolate the exponential term before taking logarithms of both sides

Wrong move:

Ignoring a negative result after isolating the exponential term

Why:

Logarithms are only defined for positive inputs, so a negative right-hand side means no solution

Correct move:

Check the sign after isolating; if negative, state there are no real solutions

Wrong move:

Mixing logarithm bases when calculating

Why:

Inconsistent bases lead to incorrect values for the unknown variable

Correct move:

Use the same logarithm base for both sides of the equation

Wrong move:

Rounding intermediate steps too early

Why:

Premature rounding leads to inaccuracies in the final answer that lose exam marks

Correct move:

Keep full calculator precision until the final step, then round to the required significant figures

5. Quick Reference Cheatsheet

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

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 $3^{x} = 10$ for x
2022 · 2
Exponential population growth problem
2023 · 1
Solve $e^{2 x} = 15$

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.