IB Maths AI SL · Unit 1: Number and Algebra
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
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 →
The question bank covers these logarithms and exponentials question types (number of questions in brackets):
Use your GDC to solve $4^x = 35$. Give your answer to 3 s.f.
1. Open the graphing app.
2. Enter $Y_1 = 4^x$.
3. Enter $Y_2 = 35$.
4. Use the intersection tool (G-Solv $\to$ ISCT).
5. Read: intersection at $x = 2.5646\ldots \implies \mathbf{x = 2.56}$.
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 $2000$.
1. Substitute $t = 6$: $P(6) = 400(1.15)^6$.
2. Evaluate: $925.22\ldots \implies \mathbf{925}$ bacteria (must be a whole number).
3. Set $P = 2000$: $2000 = 400(1.15)^t$.
4. Isolate the exponential: $5 = 1.15^t$.
5. Solve on the GDC (intersection of $Y_1 = 1.15^x$ and $Y_2 = 5$).
6. Read: $t = 11.516\ldots \implies \mathbf{11.5\text{ hours}}$.
The temperature $T$ (in $^\circ\text{C}$) of a cooling cup of tea $t$ minutes after it is poured is modelled by $T(t) = 22 + 75e^{-0.05t}$. Find the initial temperature of the tea, and determine how many minutes it takes for the tea to cool to $45^\circ\text{C}$.
1. Substitute $t = 0$: $T(0) = 22 + 75e^0$.
2. Evaluate: $e^0 = 1$, so $T(0) = 22 + 75 = \mathbf{97^\circ\text{C}}$.
3. Set $T = 45$: $45 = 22 + 75e^{-0.05t}$.
4. Rearrange: $23 = 75e^{-0.05t} \implies \frac{23}{75} = e^{-0.05t}$.
5. Solve on the GDC numerical solver.
6. Read: $t = 23.633\ldots \implies \mathbf{23.6\text{ minutes}}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Using the logarithm functions on your Graphic Display Calculator, evaluate the following expressions:
\(\log_{10} 1000\)
\(\ln(e^4)\)
\(e^{\ln 7}\)
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.
The temperature of a cup of coffee, \(T\) in degrees Celsius, \(t\) minutes after being poured is given by the exponential model \(T(t) = 20 + 75e^{-0.08t}\).
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 \(45^\circ\text{C}\).
All 16 logarithms and exponentials questions with mark schemes →
There are 16 exam-style logarithms and exponentials questions in the AI SL question bank (Paper 1: 11 · Paper 2: 5), graded 3 easy, 2 medium, 5 hard, 4 very hard, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 11 · Paper 2: 5. Practise with your GDC — AI papers expect calculator methods throughout.
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.