Combine logarithms and change their base
Apply product, quotient and power laws with valid domains, then convert a logarithm to another base.
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
Which expression equals log₅(MN)?
What you will learn
- Use the product, quotient and power laws of logarithms
- Combine logarithmic terms into one logarithm
- Use change of base to evaluate or rewrite a logarithm
Log laws turn products into sums
For a common base b > 0 with b ≠ 1 and positive arguments, log_b(MN) = log_b M + log_b N; log_b(M/N) = log_b M − log_b N; and log_b(Mʳ) = r log_b M.
These laws combine logarithms only when their bases match. A coefficient becomes an exponent when moving into a single logarithm, and a minus sign becomes division.
A standalone constant can be written as a logarithm: for example, 3 = log₁₀(1000). Always keep the original argument restrictions, such as p > 0 and q > 0.
Change base with a ratio
For positive M and bases a, b > 0 with a ≠ 1 and b ≠ 1, log_a M = log_b M / log_b a. In particular, log_a M = ln M / ln a.
On a calculator, use the same base in numerator and denominator. For an exact rewrite, keep the logarithm ratio rather than rounding it.
Natural logarithms use base e; common logarithms use base 10. Their laws are the same as for any other valid base.
Pause and explain
How can log₄ 9 be evaluated with a calculator that has ln?
Worked example
Write 3 + 2 log₁₀ p − log₁₀ q as a single base-10 logarithm.
Show the worked solution
- Rewrite 3 as log₁₀(1000).
- Use the power law: 2 log₁₀ p = log₁₀(p²).
- Combine the sum as a product and the subtraction as a quotient.
Answer log₁₀(1000p²/q), for p > 0 and q > 0.
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.
Combine log₂ x + 3 log₂ y − log₂ z into one logarithm.
Give me a hint
Turn the coefficient into a power, then combine product and quotient.
Compare my reasoning
- 3 log₂ y = log₂(y³).
- The two positive terms combine as log₂(xy³).
- Subtracting log₂ z divides by z.
log₂(xy³/z), where x, y and z are positive.
Look for these in your work
- Power law used
- Product combined
- Quotient and domain stated
Write log₅ 12 / log₅ 2 as one logarithm, then simplify it.
Give me a hint
Use the quotient law and reduce the argument.
Compare my reasoning
- A difference of logs is log₅(12/2).
- 12/2 = 6.
- The base remains 5.
log₅ 6.
Look for these in your work
- Quotient law applied
- Argument simplified
- Base retained
A sound level uses L = 10 log₁₀(I/I₀). Rewrite it using natural logarithms and state the required condition on the intensity ratio.
Give me a hint
Replace log₁₀ R by ln R / ln 10.
Compare my reasoning
- Let R = I/I₀; the logarithm requires R > 0.
- Change base: L = 10(ln R / ln 10).
- Keep ln 10 as the denominator for an exact form.
L = 10 ln(I/I₀) / ln 10, with I/I₀ > 0.
Look for these in your work
- Ratio identified
- Base conversion correct
- Domain checked
Common mistakes
- Combining terms with different bases
- Moving a coefficient without making it an exponent
- Losing a denominator when combining a subtraction
- Ignoring positivity restrictions on logarithm arguments
What can you explain now?
Rewrite log₃ 7 using natural logarithms.
Compare with the explanation
ln 7 / ln 3
Change base divides the logarithm of the argument by the logarithm of the old base.
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Make it stick
State the product, quotient, power and change-of-base laws, including their domain conditions.
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.