IB Maths: Exponential & Logarithmic Functions FAQ

Start by identifying whether you're solving, simplifying, or sketching. For equations, isolate the exponential or log term first, then apply $\ln$ or the inverse exponential to both sides. Always check your answer's domain — logs of negative numbers or zero don't exist, and examiners plant that trap constantly.

An exponential function has the form $f(x)=a^x$ where the variable is the power; a logarithmic function, $f(x)={\log }_{a}x$, is its inverse — it undoes the exponential. Graph one and reflect it in the line $y=x$ and you get the other. That relationship is the whole topic in one sentence.

You need four: ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$, ${\log }_a(x/y)={\log }_{a}x-{\log }_{a}y$, ${\log }_a(x^n)=n{\log }_{a}x$, and the change of base rule ${\log }_{a}x=\frac{\ln x}{\ln a}$. These are given in the formula booklet but you must know when to apply each.

Try to write both sides with the same base first — this avoids logs entirely and is what Paper 1 usually wants. If $9^x=27$, rewrite as $3^{2x}=3^3$, so $2x = 3$ and $x=1.5$. Only reach for logarithms when a common base isn't possible.

AI treats exponentials mostly through modelling — compound growth, depreciation, population curves — with heavier use of GDC and the finance solver. AA goes deeper into the algebra: proving log laws, working with $e$ and natural logs analytically, and differentiating/integrating exponential and log functions by hand, especially at HL.

Yes — natural log and the constant $e$ appear in the SL syllabus for both AA and AI, not just HL. HL students go further, differentiating and integrating $\ln x$ and $e^x$ and using $e$ in more complex modelling contexts, including continuous compound growth.

Expect short solve/simplify questions on Paper 1 and Paper 2, and longer modelling questions on Paper 3 (HL only) that combine logs with calculus or sequences. Command terms like "solve", "show that" and "hence" are common — "hence" means you must use your previous result, not start fresh.

The top three I mark every session: forgetting the domain restriction ($x>0$ for logs), misapplying the addition/subtraction laws to sums instead of products, and dropping the base when using change-of-base. A close fourth is forgetting that ${\log }_{a}1=0$ for any base.

Master the four log laws cold, practise solving equations both with and without a calculator, and always state domain restrictions — that's the mark students lose most. Then move to past-paper questions that combine logs with calculus or sequences, since a 7 depends on connecting topics, not just isolated recall.

Look for questions combining exponentials with sequences (geometric growth), calculus (differentiating $e^{kx}$), and financial or scientific modelling — these appear most often. Topical worksheets grouped specifically by exponential/log content, rather than mixed past papers, are the fastest way to drill weak spots before a mock.

They link directly to sequences and series (geometric growth is exponential), calculus (differentiating $e^x$ gives itself), and statistics (continuous growth/decay models). Financial maths questions on AI papers almost always use the exponential growth formula, so this topic rarely stands alone in an exam.

It's a mid-difficulty topic — the concepts themselves aren't the hardest in the course, but students who rush the log laws early on struggle badly later when logs reappear in calculus and modelling questions. Most difficulty comes from weak algebra foundations, not the topic itself.

Yes, generally — Analysis and Approaches asks students to manipulate logs algebraically and apply calculus to exponential functions, while Applications and Interpretation leans on calculator-based modelling. A student who finds pure algebra uncomfortable often finds this topic more manageable in AI.

The single most useful thing you can check is whether your child can state the four log laws from memory without looking them up — if not, that's the gap to close first. Past-paper practice on this specific topic, marked against the official IB markscheme, is more useful than general revision.