SL 1.5—Exponents and Logarithms Questionbank

Practice SL 1.5—Exponents and Logarithms with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

Consider the properties of logarithms, specifically the product and power rules.

Using the identity $\log_a(xy) = \log_a x + \log_a y$ , calculate $\log_2(32 \cdot 4)$ .

[4]

If $\log_a x = 3$ and $\log_a y = 2$ , find the value of $\log_a(x^2y^3)$ .

[3]

Question 2

HLPaper 2

The function $f(x)=\ln\left(\frac{1}{x-2}\right)$ is defined for $x>2, \ x \in \mathbb{R}$ .

Find an expression for $f^{-1}(x)$ . You are not required to state a domain.

[4]

Solve $f(x)=f^{-1}(x)$ .

[1]

Question 3

SLPaper 1

The amount, in milligrams, of a medicinal drug in the body $t$ hours after it was injected is given by $D(t) = 23(0.85)^t, \ t \geq 0$ . Before this injection, the amount of the drug in the body was zero.

Write down the initial dose of the drug.

[1]

Write down the percentage of the drug that leaves the body each hour.

[2]

Calculate the amount of the drug remaining in the body 10 hours after the injection.

[2]

Question 4

SLPaper 2

A car depreciates in value exponentially. The initial value of the car is $20,000, and it depreciates at a rate of 15% per year.

Calculate the value of the car after 5 years.

[2]

Determine the time it takes for the car's value to halve.

[2]

Question 5

HLPaper 2

Consider the equation $x^5 - 3x^4 + mx^3 + nx^2 + px + q = 0$ , where $m, n, p, q \in \mathbb{R}$ .

The equation has three distinct real roots which can be written as $\log_2 a$ , $\log_2 b$ and $\log_2 c$ .

The equation also has two imaginary roots, one of which is $di$ where $d \in \mathbb{R}$ .

The values $a, b, c$ are consecutive terms in a geometric sequence.

Show that $abc = 8$ .

[5]

Show that one of the real roots is equal to 1.

[3]

Given that $q = 8d^2$ , find the other two real roots.

[9]

Question 6

HLPaper 1

Solve the equation $4^x + 2^{x+2} = 3$ .

[6]

Question 7

HLPaper 1

The graph of the function $f(x) = \ln x$ is translated by the vector $\begin{pmatrix} a \\ b \end{pmatrix}$ so that it passes through the points $(0, 1)$ and $(e^3, 1 + \ln 2)$ .

Find the value of $a$ and the value of $b$ .

[6]

Question 8

SLPaper 1

In an experiment, a number of fruit flies are placed in a container. The population of fruit flies, $P$ , increases and can be modelled by the function

$P(t) = 12 \times 3^{0.498t}, \quad t \geqslant 0$

where $t$ is the number of days since the fruit flies were placed in the container.

Find the number of fruit flies which were placed in the container.

[2]

Find the number of fruit flies that are in the container after 6 days.

[2]

The maximum capacity of the container is 8000 fruit flies.

Find the number of days until the container reaches its maximum capacity.

[2]

Question 9

SLPaper 1

The first three terms of a geometric sequence are $\ln x^{16}$ , $\ln x^8$ , $\ln x^4$ , for $x > 0$ .

Find the common ratio.

[3]

Question 10

SLPaper 1

The speed of light is $300\ 000$ kilometres per second. The average distance from the Sun to the Earth is 149.6 million km.

Calculate the time, in minutes, it takes for light from the Sun to reach the Earth.

[3]