A piece of industrial machinery depreciates in value according to the model $V(t) = 45\,000 e^{-0.15t}$ , where $V$ is the value in euros and $t$ is the time in years since it was purchased. Calculate the value of the machinery exactly 4 years after purchase. Round your answer to the nearest euro. Using your GDC, find the exact number of years it will take for the machinery’s value to halve.

CG50: In Run-Matrix, type $45000 \times e^{-0.15 \times 4}$ (M1) $V(4) = 24696.523\dots \implies \text{€}24\,697$ A1 Half the value $= \text{€}22\,500$ , so solve $22500 = 45000e^{-0.15t}$ (M1) CG50: MENU $\rightarrow$ Equation $\rightarrow$ Solver (F3) . Enter $22500 = 45000 \times e^{-0.15x}$ . Press SOLV (F6) (M1) $t = 4.62$ years ( $4.62098\dots$ ) A1

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IB Maths AI SL · Unit 1: Number and Algebra

IB Maths AI SL Logarithms and Exponentials Questions

Updated October 2026

Exam-style IB Maths AI SL logarithms and exponentials 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.

16 questions
Paper 1: 11
Paper 2: 5
3 easy
4 medium
5 hard
4 very hard
3 worked examples

Practise Logarithms and Exponentials 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

SL AI rarely tests logs abstractly — it tests them in exponential growth/decay: bacterial cultures, radioactive decay, depreciation. Know how to solve a·b^x = c using logs. Exponents and logarithms in real-world models overview →

What's examined in AI SL logarithms and exponentials

The question bank covers these logarithms and exponentials question types (number of questions in brackets):

Exponential Modelling (9)
Solving Exp/Log Equations (4)
Logarithm Properties (3)

Key formulas

Exponents: $a^{p} \cdot a^{q} = a^{p + q}, \frac{a^{p}}{a^{q}} = a^{p - q}, (a^{p})^{q} = a^{p q}$
Logarithms — power law: $\log_{a} (x^{n}) = n \log_{a} x$

In the same notation as the IB formula booklet. All AI SL formulas →

Explained, with a worked example: Laws of logarithms

Where students lose marks on logarithms and exponentials

Examiner's noteExponential and log models: e⁰ = 1, the equation vs the inequality, reading the model

Where marks are lost: Candidates treated a term with exponent 0 as zero, rounded b to 1 in f(t) = abᵗ, misread log₁₀x in a model, set an equation where the context needed an inequality (and then rounded the wrong way), gave calculator mantissas as answers, and commented on a model without using its equation.

Examiners flagged this in 3 exam series (May 2024, November 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Logarithms and Exponentials worked examples

Worked example 1: Solving an exponential equation graphically · easy

Use your GDC to solve $4^{x} = 35$ . Give your answer to 3 s.f.

Solution

5 steps · 1 to fill in

Examiner tip: AI SL students are fully encouraged to solve exponential equations with a GDC intersection or numerical solver — you are not required to use logs analytically.

Worked example 2: Evaluating a bacterial growth model · medium

The bacteria population is modelled by $P (t) = 400 (1.15)^{t}$ , where $t$ is time in hours. Calculate the population after $6$ hours, and find the time for the population to reach $2000$ .

Solution

6 steps

Examiner tip: Context dictates rounding. Populations of living things should be rounded to the nearest whole number — you can't have

0.22

of a bacterium.

Worked example 3: Newton's Law of Cooling · hard

The temperature $T$ (in $^{\circ} C$ ) of a cooling cup of tea $t$ minutes after it is poured is modelled by $T (t) = 22 + 75 e^{- 0.05 t}$ . Find the initial temperature of the tea, and determine how many minutes it takes for the tea to cool to $45^{\circ} C$ .

Solution

6 steps

Examiner tip: The constant term (

22

) sitting outside the exponential is the horizontal asymptote — that is the ambient room temperature the tea will eventually cool towards.

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 AI SL logarithms and exponentials 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

Using the logarithm functions on your Graphic Display Calculator, evaluate the following expressions:

$\log_{10} 1000$
$\ln (e^{4})$
$e^{\ln 7}$

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

A piece of industrial machinery depreciates in value according to the model $V (t) = 45 000 e^{- 0.15 t}$ , where $V$ is the value in euros and $t$ is the time in years since it was purchased.

Calculate the value of the machinery exactly 4 years after purchase. Round your answer to the nearest euro.
Using your GDC, find the exact number of years it will take for the machinery’s value to halve.

Full worked solution Free, no sign-in

CG50: In Run-Matrix, type $45000 \times e^{- 0.15 \times 4}$ (M1)
$V (4) = 24696.523 \dots ⟹ € 24 697$ A1
Half the value $= € 22 500$ , so solve $22500 = 45000 e^{- 0.15 t}$ (M1)
CG50: MENU $\to$ Equation $\to$ Solver (F3). Enter $22500 = 45000 \times e^{- 0.15 x}$ . Press SOLV (F6) (M1)
$t = 4.62$ years ( $4.62098 \dots$ ) A1

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

Question 3 · hard · 8 marks · Paper 2

The temperature of a cup of coffee, $T$ in degrees Celsius, $t$ minutes after being poured is given by the exponential model $T (t) = 20 + 75 e^{- 0.08 t}$ .

Find the initial temperature of the coffee.
Calculate the temperature of the coffee 10 minutes after it was poured.
Write down the equation of the horizontal asymptote of the graph of $T$ , and state its meaning in the context of this problem.
Using your GDC, find the time it takes for the coffee to cool down to exactly $45^{\circ} C$ .

Attempt it and see the mark scheme →

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All 16 logarithms and exponentials questions with mark schemes →

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Revise logarithms and exponentials for the exam

FAQ

How many IB Maths AI SL logarithms and exponentials questions are there?

There are 16 exam-style logarithms and exponentials questions in the AI SL question bank (Paper 1: 11 · Paper 2: 5), graded 3 easy, 4 medium, 5 hard, 4 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is logarithms and exponentials on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 11 · Paper 2: 5. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 1 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.

What else is free here

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 AI SL Logarithms and Exponentials Questions

Try it first

What you need to know

What's examined in AI SL logarithms and exponentials

Key formulas

Where students lose marks on logarithms and exponentials

Logarithms and Exponentials worked examples

Worked example 1: Solving an exponential equation graphically · easy

Solution

Worked example 2: Evaluating a bacterial growth model · medium

Solution

Worked example 3: Newton's Law of Cooling · hard

Solution

Try these IB Maths AI SL logarithms and exponentials 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 · 8 marks · Paper 2

Revise logarithms and exponentials for the exam

FAQ

What else is free here

More AI SL Unit 1 topics