IB Maths AI HL · Unit 1: Number and Algebra
IB Maths AI HL Logs and Infinite Geometric Series Questions
Exam-style IB Maths AI HL logs and infinite geometric series 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.
- 26 questions
- Paper 1: 26
- 5 easy
- 10 medium
- 9 hard
- 2 starter
- 3 worked examples
Practise Logs and Infinite Geometric Series questions →
AI HL formula booklet
What's examined in AI HL logs and infinite geometric series
The question bank covers these logs and infinite geometric series question types (number of questions in brackets):
- Series with Logarithmic Terms (14)
- Solving Logarithmic/Exponential Equations (7)
- Geometric Series Summation (5)
Logs and Infinite Geometric Series worked examples
Worked example 1: Summing an Infinite Geometric Series · easy
An infinite geometric series has a first term of $u_1 = 45$ and a common ratio of $r = 0.92$. Calculate the exact sum of the infinite series, $S_\infty$.
1. Identify the formula for the sum to infinity of a geometric series: $S_\infty = \frac{u_1}{1 - r}$.
2. Substitute the known values into the formula: $S_\infty = \frac{45}{1 - 0.92}$.
3. Simplify the denominator: $1 - 0.92 = 0.08$.
4. Evaluate the final division: $S_\infty = \frac{45}{0.08}$.
5. The exact sum of the infinite series is $562.5$.
Examiner tip: An infinite geometric series only converges to a finite sum if the absolute value of the common ratio is strictly less than 1 ($|r| < 1$).
Worked example 2: Solving Exponential Equations with Logarithms · medium
Solve the exponential equation $200e^{-0.05t} = 40$ algebraically, giving your exact answer in the form $t = a \ln b$, where $a, b \in \mathbb{Z}^+$.
1. Isolate the exponential term by dividing both sides by $200$: $e^{-0.05t} = \frac{40}{200} = 0.2$.
2. Rewrite $0.2$ as a fraction to help with logarithm laws: $e^{-0.05t} = \frac{1}{5} = 5^{-1}$.
3. Take the natural logarithm ($\ln$) of both sides to remove the base $e$: $\ln(e^{-0.05t}) = \ln(5^{-1})$.
4. Simplify using power laws: $-0.05t = -\ln 5$.
5. Divide to isolate $t$: $t = \frac{-\ln 5}{-0.05} =$ $20 \ln 5$.
Examiner tip: Always completely isolate the base and its exponent before applying logarithms to both sides of an algebraic equation.
Worked example 3: Analysing a Logarithmic Sequence · hard
The first three terms of a sequence are given by $u_1 = \ln 2$, $u_2 = \ln 4$, and $u_3 = \ln 8$. Prove that this sequence is arithmetic and find the exact sum of the first $10$ terms, $S_{10}$.
1. Rewrite the terms using the power law for logarithms ($\ln(a^b) = b\ln a$): $u_1 = \ln 2$, $u_2 = 2\ln 2$, and $u_3 = 3\ln 2$.
2. Calculate the difference between consecutive terms: $u_2 - u_1 = \ln 2$ and $u_3 - u_2 = \ln 2$. Since the difference is constant, the sequence is arithmetic with $d = \ln 2$.
3. Identify the arithmetic series sum formula: $S_{n} = \frac{n}{2}(2u_1 + (n-1)d)$.
4. Substitute $n=10, u_1=\ln 2$, and $d=\ln 2$: $S_{10} = 5(2\ln 2 + 9\ln 2)$.
5. Simplify the brackets to find the exact sum: $S_{10} = 5(11\ln 2) =$ $55\ln 2$.
Examiner tip: Recognize that a sequence constructed by taking the logarithms of the terms of a geometric sequence will inherently form an arithmetic sequence.
Try these IB Maths AI HL logs and infinite geometric series questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 1
By using the laws of logarithms, write the following expression as a single natural logarithm. Show your working clearly.
\[\ln(100) - 2\ln(5)\]
Question 2 · medium · 5 marks · Paper 1
An infinite geometric series has a first term $u_1 = 20$. The common ratio, $r$, satisfies the equation $ \ln(r) = \ln(3) - \ln(5) $.
(a) Show that $ r = \frac{3}{5} $. [2 marks]
(b) Calculate the sum to infinity, $ S_\infty $, of this series. [3 marks]
Attempt it and see the mark scheme →
Question 3 · hard · 8 marks · Paper 1
An infinite geometric series has first term \(u_1 = \ln\left(x^{k^2}\right)\), second term \(u_2 = \ln\left(x^k\right)\) and third term \(u_3 = \ln x\), where \(x \gt 1\) and \(k\) is a non-zero integer.
Show that the common ratio is \(r = \frac{1}{k}\). [2 marks]
Determine the possible values of \(k\) for which the series converges. [2 marks]
Given that the sum to infinity, \(S_\infty\), is equal to \(8\ln x\), find the value of \(k\). [4 marks]
All 26 logs and infinite geometric series questions with mark schemes →
FAQ
How many IB Maths AI HL logs and infinite geometric series questions are there?
There are 26 exam-style logs and infinite geometric series questions in the AI HL question bank (Paper 1: 26), graded 5 easy, 10 medium, 9 hard, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is logs and infinite geometric series on Paper 1 or Paper 2?
In the question bank these questions are set as Paper 1 questions.
Where can I get the mark schemes?
Open the AI HL 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.
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