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IB Maths AA SL · Unit 2: Functions

IB Maths AA SL Exponential and Logarithmic Functions Questions: graphs and models

Updated October 2026

Exam-style IB Maths AA SL exponential and logarithmic functions questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

25 questions
Paper 1: 16
Paper 2: 9
7 easy
9 medium
7 hard
2 very hard
3 worked examples

Practise Exponential and Logarithmic Functions questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What you need to know

y = e^x and y = ln(x): their graphs, the laws of logarithms and how to solve equations involving both. Non-calculator Paper 1 expects you to know ln e = 1 and ln 1 = 0. Exponential and logarithmic functions overview →

What's examined in AA SL exponential and logarithmic functions

The question bank covers these exponential and logarithmic functions question types (number of questions in brackets):

Modelling and Graphing Functions (15)
Solving Exp Log Equations (7)
Log Laws and Function Properties (3)

Exponential and Logarithmic Functions worked examples

Worked example 1: Solving basic exponential equations · easy

Solve $e^{3 x - 1} = 20$ algebraically. Give your exact answer in terms of natural logarithms.

Solution

4 steps

Examiner tip: Do not approximate

\ln 20

with a decimal when a question specifically asks for an exact algebraic answer. Doing so will cost you the final accuracy mark.

Worked example 2: Logarithmic laws combination · medium

Solve $\log_{2} x + \log_{2} (x - 6) = 4$ algebraically.

Solution

5 steps · 1 to fill in

Examiner tip: Always check final roots against the domain of the original logarithms; arguments must be strictly positive. Leaving

x = - 2

as a final answer will lose reasoning marks.

Worked example 3: Hidden quadratics in exponentials · hard

Solve $e^{2 x} - 8 e^{x} + 15 = 0$ algebraically. Give exact answers in logarithmic form.

Solution

5 steps · 1 to fill in

Examiner tip: Using a substitution

u = e^{x}

makes the quadratic structure obvious and drastically reduces factorisation errors.

Play with it Exponential growth model: drag the sliders

The table shows the number of bacteria (in thousands) after x hours. Change a and b so that y = a·bˣ fits the data (the check wants R² of at least 0.99). What do a and b mean in this context?

Want more? Open the full Grapher.

Try these IB Maths AA SL exponential and logarithmic functions questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 4 marks · Paper 1

Solve the following equations exactly:

$e^{3 x - 1} = 20$
$\ln (2 x) = 5$

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

Solve the exponential equation analytically by expressing both sides with a common base: $9^{x - 2} = {(\frac{1}{27})}^{x + 1}$

Full worked solution Free, no sign-in

Convert both sides to a common base of $3$ . (M1)
$9^{x - 2} = (3^{2})^{x - 2} = 3^{2 x - 4}$ . (A1)
${(\frac{1}{27})}^{x + 1} = (3^{- 3})^{x + 1} = 3^{- 3 x - 3}$ . (A1)
Equate the exponents: $2 x - 4 = - 3 x - 3$ . (M1)
$5 x = 1 ⟹ x = 0.2$ (or $\frac{1}{5}$ ). A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 1

Use the change of base formula to solve the following equation analytically: $\log_{4} x + \log_{16} x = 3$

Attempt it and see the mark scheme →

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All 25 exponential and logarithmic functions questions with mark schemes →

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FAQ

How many IB Maths AA SL exponential and logarithmic functions questions are there?

There are 25 exam-style exponential and logarithmic functions questions in the AA SL question bank (Paper 1: 16 · Paper 2: 9), graded 7 easy, 9 medium, 7 hard, 2 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is exponential and logarithmic functions on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 16 · Paper 2: 9. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA SL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

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Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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IB Maths AA SL Exponential and Logarithmic Functions Questions: graphs and models

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What you need to know

What's examined in AA SL exponential and logarithmic functions

Exponential and Logarithmic Functions worked examples

Worked example 1: Solving basic exponential equations · easy

Solution

Worked example 2: Logarithmic laws combination · medium

Solution

Worked example 3: Hidden quadratics in exponentials · hard

Solution

Try these IB Maths AA SL exponential and logarithmic functions questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 1

Revise exponential and logarithmic functions for the exam

FAQ

What else is free here

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