Exponentials and logarithms are small topics on the syllabus list but they are everywhere in IB maths: in growth and decay models, in solving equations, in calculus with eˣ and ln x, and in statistics and finance. A student who is shaky with logs loses marks in places that don't look like "log questions" at all.
The core ideas
- An exponential has the variable in the power: aˣ, eˣ.
- A logarithm answers "what power?": log_a b = c means aᶜ = b.
- e is the natural base; ln x means log base e.
- Exponentials and logs are inverse functions – their graphs are reflections in y = x.
The laws (AA)
- log a + log b = log(ab)
- log a − log b = log(a/b)
- n log a = log(aⁿ)
- Change of base: log_a b = log b / log a
- log_a a = 1 and log_a 1 = 0
And just as important – what is not true: log(a + b) is not log a + log b, and (log a)² is not 2 log a.
Solving exponential equations exactly
For AA Paper 1, answers are usually required in exact form:
- Same base: rewrite both sides with the same base and equate powers (4ˣ = 8 → 2²ˣ = 2³ → x = 3/2).
- Take logs of both sides: 3ˣ = 7 → x = ln 7 / ln 3.
- Quadratic in disguise: e²ˣ − 5eˣ + 6 = 0 → let y = eˣ, solve, then check that each solution is positive (eˣ can't be negative or zero).
AA vs AI
- AA SL and HL: the full log laws, change of base, exact solutions, and graphs of exponential and log functions with transformations. Logs then feed into calculus (derivatives of eˣ and ln x).
- AI SL: logarithms as a tool for solving exponential equations, mostly with the GDC, and exponential models for growth and decay. In the new course (first exams May 2029) logarithms leave AI SL – current students still need them.
- AI HL: also uses logs to linearise data (log–log and semi-log graphs) when fitting models.
The classic errors
- Splitting log(a + b).
- Forgetting that you can't take the log of zero or a negative number – and not rejecting impossible solutions.
- Dropping the base in change-of-base calculations.
- Writing decimals in Paper 1 when exact answers are expected.
- Confusing eˣ graphs (asymptote y = 0) with ln x graphs (asymptote x = 0).
How to practise
- Ten minutes a day: three log simplifications, two exact exponential equations.
- Sketch y = eˣ, y = ln x and one transformed version each day for a week.
- Mix with calculus: derivatives and integrals involving eˣ and ln x.
- For AI, practise exponential models from context – and the "interpret" sentence.
How Maths Bodhi helps
- A short diagnostic of index laws – most log problems are really index problems.
- Exact-answer practice for AA Paper 1.
- Model-based practice for AI, with the GDC and interpretation.
- Past IB questions marked against the markscheme.
Ajay Vatsyayan (M.Tech, 14+ years teaching maths) teaches IB DP maths at Maths Bodhi and has taught students from schools across Gurgaon, including Pathways World School, Scottish High International School, Lancers International School and DPS International. Maths Bodhi is independent and not affiliated with any school. Read Ajay's profile, or see our IB algebra tutor and IB functions tutor pages.
Home sessions across Gurgaon or online. Book a free demo class.
Related: AA functions and transformations, AA Paper 1 without a calculator, sequences, series and financial maths and the IB DP maths guide for Gurgaon families.
Course content checked against the current IB AA and AI guides and the 2029 update notes by the Maths Bodhi team (October 2026). Reviewed by Ajay Vatsyayan.
Frequently asked questions
- What are the log laws in IB maths?
- log a + log b = log(ab), log a − log b = log(a/b), n log a = log(aⁿ), and the change of base formula log_a b = log b / log a.
- How do I solve an exponential equation exactly?
- Rewrite both sides with the same base and equate powers, or take logs of both sides – for example 3ˣ = 7 gives x = ln 7 / ln 3.
- Are logarithms in IB maths AI SL?
- Yes, for exams in 2027 and 2028. They are removed from AI SL in the new course with first exams in May 2029.