HL AA · Functions
Logarithm laws and change of base (IB Math AA HL)
Updated October 2026
Quick answer: The laws: log(ab) = log a + log b, log(a/b) = log a − log b and log(aⁿ) = n log a. Change of base: logₐ x = log_b x ÷ log_b a. Use change of base to write every logarithm in one base before solving, and take logs of both sides to bring down an unknown power.
log_a(x) = ln(x)/ln(a). HL AA Paper 1 tests all three log laws in combination — practise until you can spot the shortcut.
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Examiner Insights Examiners say: Logs and exponentials: hidden quadratics and misapplied log laws. May 2024: candidates who did not spot a hidden quadratic took logs of a sum. November 2025: log(10x)³ was misread, so the power was applied to only part of the argument. The SL reports show the same log-law errors every session. What the IB subject reports say →
Key ideas
logₐ x = y means aʸ = x, so logₐ a = 1 and logₐ 1 = 0. Change of base turns log₄ x into log₂ x ÷ log₂ 4 = ½ log₂ x. It also gives logₐ b = 1/log_b a. An equation with the unknown in a power: take ln of both sides, use the power law, then collect the terms in x. There is no law for log(a + b). Do not split a logarithm of a sum. Check each answer in the original equation: a logarithm of a negative number or zero is not defined, so some solutions must be rejected. Worked example
(a) Solve log₂ x + log₄ x = 6. (b) Solve 2ˣ⁺¹ = 3ˣ, giving x in the form ln a ÷ ln b.
(a) Change of base: log₄ x = log₂ x ÷ log₂ 4 = ½ log₂ x. So log₂ x + ½ log₂ x = 6, which gives (3/2) log₂ x = 6 and log₂ x = 4. x = 2⁴ = 16. Check: log₂ 16 + log₄ 16 = 4 + 2 = 6. (b) Take ln of both sides: (x + 1) ln 2 = x ln 3. x ln 2 + ln 2 = x ln 3, so x(ln 3 − ln 2) = ln 2. x = ln 2 ÷ (ln 3 − ln 2) = ln 2 ÷ ln 1.5 ≈ 1.71. Answer: (a) x = 16. (b) x = ln 2 ÷ ln 1.5 ≈ 1.71.
Exam-style HL AA questions
Question 1 of 5 Question 2 of 5 Question 3 of 5 Question 4 of 5 Question 5 of 5 5 quick questions on logarithms. No sign-in needed.
Solve \(\log_{3} x = 4\).
Solve \(4^x = 30\). Give your answer to 3 significant figures.
Evaluate \(\log_{2} 32\).
Solve \(32^x = 24\). Give your answer to 3 significant figures.
Evaluate \(\log_{2} 8 + \log_{2} 4\).
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Maths IA ideas that use logarithm laws and change of base
FAQ
How do you change the base of a logarithm? Use logₐ x = log_b x ÷ log_b a for any convenient base b. For example, log₈ x = log₂ x ÷ 3, and on a calculator log₅ 20 = ln 20 ÷ ln 5.
Can I use my calculator for logarithm laws and change of base? In HL AA, Paper 1 is non-calculator and the other papers allow a GDC, so learn the exact method as well as the calculator one.
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