Solve equations with an unknown exponent
Solve aˣ = b by matching bases or taking logarithms, and check that the solution respects the equation's domain.
Section 6 · Logarithmic and exponential functions
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
When can equal exponents be inferred from aˣ = aʳ?
What you will learn
- Solve aˣ = b when equal bases make the exponent clear
- Use logarithms to solve when bases cannot be matched
- Check positive bases and right-hand sides before solving
First check whether the bases can be matched
For a > 0 and a ≠ 1, aˣ is positive for every real x. Therefore aˣ = b has no real solution when b ≤ 0.
If both sides can be expressed as powers of the same positive base, equate the exponents. For example, 8ˣ = 2⁹ becomes 2³ˣ = 2⁹, so x = 3.
When the bases cannot be matched conveniently, take natural logarithms of both positive sides. Then x ln a = ln b, so x = ln b / ln a.
Check the solution in the original equation
For a > 1, aˣ increases with x; for 0 < a < 1, it decreases. In either case there is one real solution for each b > 0.
If the exponent is a linear expression, isolate it after taking logs. Keep brackets: a^(mx+c) = b gives mx + c = ln b / ln a.
An exact logarithmic answer is often preferable. If a decimal is requested, round only at the end and substitute it back to check the scale.
Pause and explain
Can 2ˣ = 0 have a real solution?
Worked example
Solve 3ˣ = 20, giving an exact answer and a value to 3 significant figures.
Show the worked solution
- Both sides are positive, so take natural logarithms.
- x ln 3 = ln 20, hence x = ln 20 / ln 3.
- Evaluating the ratio gives x ≈ 2.73; 3²·⁷³ is approximately 20.
Answer x = ln 20 / ln 3 ≈ 2.73.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Solve 4ˣ = 11 exactly using logarithms.
Give me a hint
Take ln of both positive sides and divide by ln 4.
Compare my reasoning
- ln(4ˣ) = ln 11.
- Use ln(4ˣ) = x ln 4.
- Divide by ln 4.
x = ln 11 / ln 4.
Look for these in your work
- Logs applied to both sides
- Power rule used
- Correct ratio stated
Solve 2^(x+1) = 7, giving an exact expression for x.
Give me a hint
Take logs, then isolate x from x + 1.
Compare my reasoning
- (x + 1) ln 2 = ln 7.
- Divide by ln 2 to get x + 1 = ln 7 / ln 2.
- Subtract 1.
x = (ln 7 / ln 2) − 1.
Look for these in your work
- Exponent kept in brackets
- Log equation correct
- Constant isolated
A population model is P(t) = 500(1.08)ᵗ. After how many years does it first reach 800? Give an exact equation and a value to 3 significant figures.
Give me a hint
Divide by 500, take natural logarithms, then divide by ln 1.08.
Compare my reasoning
- Set 500(1.08)ᵗ = 800, so (1.08)ᵗ = 1.6.
- Take logs: t ln 1.08 = ln 1.6.
- Thus t = ln 1.6 / ln 1.08 ≈ 6.11 years.
t = ln 1.6 / ln 1.08 ≈ 6.11 years.
Look for these in your work
- Model equation formed
- Log solution derived
- Approximation checked
Common mistakes
- Taking a logarithm of a non-positive right-hand side
- Reversing ln b / ln a
- Rounding before finishing the calculation
- Equating exponents before expressing both sides with the same base
What can you explain now?
What is the exact solution of 5ˣ = 17?
Compare with the explanation
x = ln 17 / ln 5
Taking natural logs gives x ln 5 = ln 17.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
When should you match bases, and when should you use logarithms to solve aˣ = b?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.