Use logarithms as inverse exponential functions
Translate between exponential and logarithmic forms and solve exponential equations in pure and contextual settings.
Section 3 · Functions
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does log₂(8) equal?
What you will learn
- Understand logarithms as inverse exponentials
- Convert a^x = y to x = log_a(y)
- Use base-10 logarithms to solve exponential equations
A logarithm asks for an exponent
log_a(y) = x means a^x = y. For real values, a must be positive and not 1, and y must be positive.
The syllabus uses base 10 unless another base is stated. Change of base gives log_a(b) = log(b)/log(a), which a base-10 calculator can evaluate.
Solve for the exponent and interpret it
For a^x = b, take logarithms: x = log(b)/log(a). In a model A = A₀rᵗ, divide by A₀ before solving for t.
Check that the logarithm's input is positive and that the growth or decay multiplier matches the context. Keep full precision until the final rounding.
Pause and explain
Solve 10^x = 100.
Worked example
Solve 2^x = 10 to 3 significant figures.
Show the worked solution
- Take base-10 logarithms: x log(2) = log(10).
- Divide by log(2): x = log(10)/log(2).
- Evaluate and round at the end.
Answer x ≈ 3.32.
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 3^x = 20 to 3 significant figures.
Give me a hint
Use x = log(20)/log(3).
Compare my reasoning
- Take base-10 logs of both sides.
- x = log(20)/log(3).
- This is approximately 2.7268, or 2.73 to 3 significant figures.
x ≈ 2.73.
Look for these in your work
- Logs applied to both sides
- Change of base correct
- Rounding instruction followed
A balance follows 500(1.04)^t = 700. Solve for t in years to 3 significant figures.
Give me a hint
Divide by 500 before taking logarithms.
Compare my reasoning
- (1.04)^t = 1.4.
- t = log(1.4)/log(1.04).
- t ≈ 8.58 years.
Approximately 8.58 years.
Look for these in your work
- Starting amount divided out
- Growth factor correct
- Time rounded appropriately
A medicine amount follows A = 80(0.75)^t mg. When does it first fall below 20 mg? Give a continuous-time estimate.
Give me a hint
Solve 0.75^t < 0.25. The logarithm of 0.75 is negative, so dividing reverses the inequality.
Compare my reasoning
- Take logs: t log(0.75) < log(0.25).
- Divide by log(0.75) and reverse: t > log(0.25)/log(0.75).
- The threshold is about 4.82 time units; it is below 20 after that time.
After approximately 4.82 time units.
Look for these in your work
- Decay ratio set up
- Negative divisor reverses the inequality
- Threshold interpreted as time
Common mistakes
- Treating log as multiplication
- Taking a real logarithm of a non-positive number
- Reversing the change-of-base ratio
- Rounding before completing the calculation
What can you explain now?
Write 10³ = 1000 in logarithmic form.
Compare with the explanation
log₁₀(1000) = 3
The logarithm is the exponent that gives 1000 from base 10.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How does a logarithm reverse an exponential, and why can dividing a logarithmic inequality reverse its direction?
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.