1.5 Exponents and introduction to logarithms questions for IB Maths AA

43 questions on Exponents and introduction to logarithms, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
Part of 1 Number and algebra
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	24	5	29
Paper 2	14	0	14
Every paper	38	5	43

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AA-00694Paper 1 · 7 marks

Consider the infinite series

\log_{2} x + \log_{4} x + \log_{16} x + \log_{256} x + \dots, where x > 1,

in which the base of each logarithm is the square of the base of the previous logarithm.

(a)
Show that $\log_{4} x = \frac{1}{2} \log_{2} x$ .
2 marks
(b)
Hence, show that the series is geometric, and write down its common ratio.
2 marks
(c)
The sum to infinity of the series is $12$ . Find the value of $x$ .
3 marks

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Question 2 · AA-00533Paper 2 · 16 marks

A company releases a new mobile game. The following table shows the number of downloads, $y$ thousand, in week $t$ after the game's release.

Week ( $t$ )	1	2	3	4	5	6	7
Downloads ( $y$ thousand)	4.6	6.2	9.3	12.4	18.6	25.1	37.4

(a)
The regression line of $y$ on $t$ has equation $y = a t + b$ . Find the value of $a$ and the value of $b$ .
2 marks
(b)
Write down the value of $r$ , Pearson's product-moment correlation coefficient, for these data.
1 mark
(c)
A scatter diagram suggests that the relationship between $t$ and $y$ is not linear. Let $z = \ln y$ .
The regression line of $z$ on $t$ has equation $z = c t + d$ . Find the value of $c$ and the value of $d$ .
2 marks
(d)
Find the value of Pearson's product-moment correlation coefficient for $t$ and $z$ . Hence state which of the two regression lines gives the better model for the data.
2 marks
(e)
Show that the equation $z = c t + d$ can be written in the form $y = A e^{k t}$ , and find the value of $A$ and the value of $k$ .
3 marks
(f)
Use the model $y = A e^{k t}$ to estimate the number of downloads in week 9.
2 marks
(g)
Comment on the reliability of your estimate in part (f).
1 mark
(h)
According to the model $y = A e^{k t}$ , find the first week in which the number of downloads is more than 100 thousand.
3 marks

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Question 3 · AA-00703Paper 1 · 4 marks

Find the term in $x^{5}$ in the expansion of ${(4 - \frac{x}{2})}^{9}$ .

4 marks

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Question 4 · AA-00637Paper 2 · 5 marks

Consider the curve $y = 2^{x}$ , where $x \in R$ . The point $P (1, 2)$ lies on the curve.

The point $Q (1 + h, 2^{1 + h})$ , where $h > 0$ , also lies on the curve. The gradient of the chord $[P Q]$ is $m$ .

The following table shows some values of $h$ and the corresponding values of $m$ , correct to five significant figures.

$h$	$1$	$0.1$	$0.01$	$0.001$
$m$	$2.0000$	$1.4355$	$p$	$q$

(a)
Use your GDC to find $p$ and $q$ .
2 marks
(b)
Hence write down an estimate for the gradient of the tangent to the curve at P.
1 mark
(c)
Use your answer to part (b) to find the equation of the tangent to the curve at P. Give your answer in the form $y = a x + b$ .
2 marks

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Question 5 · AA-00707Paper 1 · 5 marks

Consider the expansion of ${(a x - \frac{3}{x})}^{7}$ , where $a \in R$ , $a \neq 0$ . The coefficient of $x^{3}$ is $\frac{189}{32}$ .

Find the value of $a$ .

5 marks

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Question 6 · AA-00650Paper 2 · 5 marks

Kofi invests $1000 in an account that pays a nominal annual interest rate of $5 %$ , compounded $n$ times per year. The value of his investment after one year, in dollars, is

A (n) = 1000 {(1 + \frac{0.05}{n})}^{n} .

The following table shows some values of $A (n)$ , correct to two decimal places.

$n$	$1$	$4$	$12$	$365$	$8760$
$A (n)$	$1050.00$	$1050.95$	$p$	$q$	$r$

(a)
Find the value of $p$ , the value of $q$ and the value of $r$ .
2 marks
(b)
Use the table to write down an estimate of $\lim_{n \to \infty} A (n)$ , correct to two decimal places.
1 mark
(c)
A second account pays a nominal annual interest rate of $5.13 %$ , compounded once per year. Kofi claims that if the first account compounded interest often enough, it would give him more money after one year than the second account.
Use your answer to part (b) to determine whether Kofi is correct.
2 marks

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