Exponents & Logs - IB Math AA Question Bank

Question 1

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

A particle has mass $4.8\times 10^{-23}\,\text{kg}$ . A sample contains $2.5\times 10^{18}$ identical particles.

A

Find the mass of the sample, in kg, giving your answer in the form $a\times 10^k$ , where $1\le a<10$ and $k\in\mathbb{Z}$ .

[2]

B

Another sample has mass $3.6\times 10^{-2}\,\text{kg}$ . Find how many times as large this mass is as the mass found in part (a). Give your answer in standard form.

[3]

Question 2

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

Let $x>0$ and

E=\frac{(2x^{-3})^2(8x^5)}{4x^{-2}}

A

Simplify $E$ in the form $ax^b$ , where $a,b\in\mathbb{Z}$ .

[3]

B

Hence solve $E=24$ .

[2]

Question 3

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

Let $A$ and $B$ be positive real numbers such that $\log A=2.37$ and $\ln B=1.18$ , where $\log$ denotes the logarithm to base $10$ .

A

Find the value of $A$ .

[2]

B

Find the value of $B$ .

[2]

C

Find $\log\left(\dfrac{A^2}{\sqrt{B}}\right)$ .

[2]

Question 4

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $a=\log_{10}2$ and $b=\log_{10}5$ .

A

Show that $a+b=1$ .

[2]

B

Write $\log_{10}40$ in terms of $a$ only.

[2]

C

Find $\log_4 100$ in terms of $a$ .

[2]

Question 5

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $x>0$ .

A

Express $2\log_3 x-\log_3 4$ as a single logarithm.

[2]

B

Hence solve $2\log_3 x-\log_3 4=2$ .

[3]

Question 6

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $x>0$ and

F=\frac{(x^{3/2})^4}{x^2\sqrt{x}}

A

Simplify $F$ as a single power of $x$ .

[3]

B

Given that $F=128$ , find $x$ .

[2]

Question 7

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $a>0$ , $a\ne1$ and $x>0$ .

A

Show that

\log_{a^2}x+\log_{\sqrt a}x=\frac{5}{2}\log_a x

[3]

B

Hence solve $\log_9x+\log_{\sqrt3}x=10$ .

[2]

Question 8

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A single bacterium has mass $7.40\times 10^{-13}\ \text{g}$ . A sample of these bacteria has total mass $0.0360\ \text{g}$ . The number of bacteria in the sample is denoted by $N_0$ . After the sample is placed in a nutrient solution, the number of bacteria is modelled by

N(t)=N_0 2^{\frac{t}{25}}

where $t$ is the time in minutes.

A

Find $N_0$ , giving your answer in the form $a\times 10^k$ , where $1\leq a<10$ and $k\in\mathbb{Z}$ .

[2]

B

Determine the time taken for the number of bacteria to reach $1.00\times 10^{12}$ .

[3]

Question 9

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

For $x>0$ , define

f(x)=\frac{(16x^8)^{\frac{3}{4}}}{2x^{\frac{1}{2}}}

A

Simplify $f(x)$ in the form $ax^p$ , where $a,p\in\mathbb{R}$ .

[3]

B

Solve $f(x)=75$ .

[2]

Question 10

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Consider the equation

\log_3(x-1)+\log_3(x+5)=2

where $x>1$ .

A

Show that this equation can be written as $x^2+4x-14=0$ .

[2]

B

Hence solve the original equation, giving the exact value of $x$ .

[3]

C

Find $\log_3(x+5)$ for this value of $x$ , giving your answer to 3 significant figures.

[1]

Question 11

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A hot drink is placed in a room. The difference, $D$ , between the temperature of the drink and the room temperature is modelled by

D(t)=48(0.82)^t

where $t$ is the time in hours and $D$ is measured in degrees Celsius.

A

Find $D(3)$ .

[2]

B

Determine the time when $D(t)=10$ .

[3]

C

State the first whole number of hours after which $D(t)$ is less than $10$ .

[1]

Question 12

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The amount, $A$ , of a radioactive substance remaining after $t$ hours is modelled by

A(t)=Ce^{-kt}

where $C$ and $k$ are positive constants. Initially there are $8.00\times 10^{-3}\ \text{mol}$ of the substance. After $12$ hours there are $2.30\times 10^{-3}\ \text{mol}$ remaining.

A

Write down the value of $C$ .

[1]

B

Find the value of $k$ .

[2]

C

Determine the time when $5.00\times 10^{-4}\ \text{mol}$ remains.

[2]

D

Find $A(30)$ , giving your answer in scientific notation.

[1]

Question 13

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A

Solve the equation $3^{2x-1}=5^{x+1}$ , giving your answer in exact form.

[4]

Question 14

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the equation

4^x-9\cdot2^x+18=0

A

Let $t=2^x$ . Write the equation as a quadratic equation in $t$ .

[2]

B

Hence solve the original equation, giving your answers in exact form.

[3]

Question 15

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $x>0$ , $x\ne1$ . Consider the equation

\log_x16+\log_4x=3

A

By using the substitution $t=\log_4x$ , show that $t^2-3t+2=0$ .

[3]

B

Hence find all possible values of $x$ .

[3]

Question 16

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $a,b,c>0$ , with $a\ne1$ and $b\ne1$ .

A

Prove that

\log_a b\cdot\log_b c=\log_a c

[3]

B

Hence evaluate

\log_2 3\cdot\log_3 5\cdot\log_5 7\cdot\log_7 32

[2]

Question 17

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the inequality

\left(\frac12\right)^{x^2-3x}<4^{x-1}

A

Rewrite the inequality using powers of $2$ only.

[2]

B

Hence solve the inequality.

[3]

Question 18

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The sound level, $L$ , in decibels, is given by

L=10\log\left(\frac{I}{I_0}\right)

where $I$ is the sound intensity and $I_0=1.00\times 10^{-12}\ \text{W m}^{-2}$ .

A

Find $L$ when $I=3.60\times 10^{-7}\ \text{W m}^{-2}$ .

[2]

B

Sound A is $18.0\ \text{dB}$ louder than sound B. Find the ratio of the intensity of sound A to the intensity of sound B.

[2]

C

Two identical machines each produce intensity $I$ . Find the increase in sound level when both machines operate together instead of one machine operating alone.

[1]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the equation

2^x+e^{-x}=4

A

Solve the equation, giving both solutions correct to three significant figures.

[3]

B

Solve $2^x=e^{-x}$ .

[1]

C

Hence, or otherwise, solve the inequality $2^x+e^{-x}<4$ .

[2]

Question 20

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the equation

\log_2 x+\log_x 16=5

where $x>0$ and $x\neq 1$ .

A

By letting $u=\log_2 x$ , show that $u^2-5u+4=0$ .

[3]

B

Hence solve the original equation.

[2]

Question 21

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the equation

\log_4(3x-1)=\log_2(x+1)-1

A

Write the equation using logarithms to base $2$ only.

[1]

B

Show that the equation can be written as $x^2-10x+5=0$ .

[2]

C

Hence solve the original equation.

[2]

Question 22

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the equation

(9^x)^{\frac{1}{2}}+3^{1-x}=7

A

By letting $u=3^x$ , show that $u^2-7u+3=0$ .

[3]

B

Hence solve the original equation, giving your answers correct to three significant figures.

[3]

Question 23

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A thin filter sheet is made from identical fibres. Each fibre has mass $3.2\times 10^{-14}\ \text{kg}$ . One layer of the sheet contains $2.5\times 10^9$ fibres, and each sheet contains $8.0\times 10^2$ layers.

A

I.

Find the mass of one layer, giving your answer in the form $a\times 10^k$ , where $1\leq a<10$ and $k\in\mathbb{Z}$ .

[2]

II.

Find the mass of one sheet, giving your answer in the form $a\times 10^k$ , where $1\leq a<10$ and $k\in\mathbb{Z}$ .

[2]

B

A container holds $1.28\times 10^3\ \text{kg}$ of these sheets. Find the number of sheets in the container.

[2]

C

I.

Each sheet has area $4.0\times 10^{-3}\ \text{m}^2$ . Find the total area of the sheets in the container.

[1]

II.

Find the fractional increase of this area over $75\ \text{m}^2$ .

[2]

Question 24

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $x>0$ and define

E=\frac{(27x^6)^{\frac{2}{3}}}{3x^{\frac{1}{2}}}

A

I.

Simplify $E$ in the form $ax^p$ , where $a,p\in\mathbb{R}$ .

[3]

II.

Hence solve $E=24\sqrt{2}$ . If you did not obtain $E=3x^{\frac{7}{2}}$ , use this expression.

[2]

B

I.

Show that $E^2=9x^7$ . If you did not obtain $E=3x^{\frac{7}{2}}$ , use this expression.

[2]

II.

Hence solve $E=\dfrac{12}{x}$ , giving your answer in exact form.

[3]

Question 25

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $a=\log_2 5$ .

A

AI.

Express $\log_2 20$ in terms of $a$ .

[1]

AII.

Express $\log_4 5$ in terms of $a$ .

[1]

AIII.

Express $\log_{10}2$ in terms of $a$ .

[2]

B

BI.

Solve $5^x=8$ , giving your answer in terms of $a$ .

[2]

BII.

Solve $20^y=50$ , giving your answer in terms of $a$ .

[2]

C

Show that $0<a<3$ .

[2]

Question 26

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The pH of a solution is defined by

\text{pH}=-\log H

where $H$ is the hydrogen ion concentration in $\text{mol dm}^{-3}$ and $\log$ denotes logarithm to base $10$ .

A

I.

Find the pH when $H=10^{-3}$ .

[1]

II.

A solution has pH $7.4$ . Express its value of $H$ in the form $10^q$ .

[2]

B

Solution B has hydrogen ion concentration $100$ times that of solution A, where solution A has pH $3$ . Find the pH of solution B.

[2]

C

I.

Equal volumes of two solutions, with concentrations $10^{-3}$ and $10^{-5}$ , are mixed. Find the concentration of the mixture in scientific notation.

[2]

II.

Find the pH of the mixture, leaving your answer in exact logarithmic form.

[2]

Question 27

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $x>0$ . Consider the equation

\log_2(8x)\log_2\left(\frac{x}{2}\right)=15

A

By using the substitution $u=\log_2x$ , show that this equation may be written as $u^2+2u-18=0$ .

[4]

B

Hence solve the original equation, giving your answers in exact form.

[3]

Question 28

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A space probe transmits a signal with power $P_0=7.20\times10^{15}\ \text{W}$ . At a distance $r$ metres from the probe, the intensity $I$ of the signal is modelled by

I=\frac{P}{4\pi r^2}

where $P$ is the transmitted power in watts.

A

I.

Find the intensity of the signal when $r=3.00\times10^{11}$ metres.

[2]

II.

The receiving station can detect signals with intensity at least $5.00\times10^{-10}\ \text{W m}^{-2}$ . Determine the greatest distance from which the signal can be detected, assuming $P=P_0$ . Give your answer in scientific notation.

[2]

B

Treat $t$ as continuous. Determine the initial intensity at this distance and the time at which the intensity reaches the detection limit. Hence state the cutoff time after which the signal is undetectable.

[3]

C

For a different receiving station, the detection limit is reduced by a factor of $100$ . Determine how this changes the greatest detectable distance, assuming the same transmitted power $P_0$ .

[3]

Question 29

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The pH of an aqueous solution is defined by

\text{pH}=-\log[\text{H}^+]

where $[\text{H}^+]$ is the hydrogen ion concentration in $\text{mol dm}^{-3}$ and $\log$ denotes logarithm to base $10$ .

A

I.

Solution A has $[\text{H}^+]=3.20\times10^{-4}\ \text{mol dm}^{-3}$ . Find its pH.

[2]

II.

Solution B has pH $4.20$ . Find $[\text{H}^+]$ for solution B.

[1]

B

I.

A $25.0\ \text{cm}^3$ sample of solution A is diluted to a total volume of $500\ \text{cm}^3$ . Find the new hydrogen ion concentration.

[2]

II.

Find the pH of the diluted solution.

[2]

C

Water is added to $40.0\ \text{cm}^3$ of solution A until the pH is $5.00$ . Determine the volume of water added.

[3]

Question 30

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

For $x>0$ , define

f(x)=\frac{(81x^{12})^{\frac14}}{3x^{-\frac12}}

A

I.

Show that $f(x)=x^{\frac72}$ .

[2]

II.

Find $f(1.8)$ .

[1]

B

Solve $f(x)=90$ , giving your answer correct to three significant figures.

[3]

C

Solve $\log_2(f(x))-\log_2(x+1)=3$ for $x>0$ .

[3]

Question 31

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The magnitude $M$ of an earthquake is modelled by

M=\log\left(\frac{A}{A_0}\right)

where $A$ is the amplitude of the seismic wave, $A_0$ is a reference amplitude, and $\log$ denotes logarithm to base $10$ .

A

I.

An earthquake has magnitude $5.80$ . Find $\frac{A}{A_0}$ .

[1]

II.

A second earthquake has magnitude $4.50$ . Find the ratio of the amplitude of the first earthquake to the amplitude of the second.

[2]

B

Two seismic waves arrive at the same station at the same time. Their individual magnitudes are $5.80$ and $4.50$ , and their amplitudes add. Find the magnitude of the combined wave.

[3]

C

After the first earthquake, the amplitude at a station is modelled by $A(t)=A_1e^{-0.18t}$ , where $t$ is measured in minutes and $A_1$ is the initial amplitude corresponding to magnitude $5.80$ . Determine the time at which the magnitude reaches $3.20$ , and hence state when it is less than $3.20$ .

[4]

Question 32

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $a>0$ , $a\neq 1$ . It is given that

\log_a12\approx2.54\quad\text{and}\quad \log_a5\approx1.65

A

Find $\log_a\left(\dfrac{144}{25}\right)$ .

[2]

B

Find the value of $a$ .

[2]

C

Solve $a^{2x}-17a^x+60=0$ .

[3]

Question 33

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

A data archive stores images. Each uncompressed image has size $S$ bytes. The archive uses binary prefixes, so $1\ \text{GiB}=2^{30}$ bytes. The number of images stored after $n$ months is modelled by

N_n=N_0(1.18)^n

$n$	$N_n$	Storage [bytes]	Capacity [bytes]
0	$2.50 \times 10^{5}$	$1.20 \times 10^{12}$	$8.00 \times 10^{12}$
1	$2.95 \times 10^{5}$	$1.42 \times 10^{12}$	$8.00 \times 10^{12}$
2	$3.48 \times 10^{5}$	$1.67 \times 10^{12}$	$8.00 \times 10^{12}$
3	$4.11 \times 10^{5}$	$1.97 \times 10^{12}$	$8.00 \times 10^{12}$
4	$4.85 \times 10^{5}$	$2.33 \times 10^{12}$	$8.00 \times 10^{12}$
5	$5.72 \times 10^{5}$	$2.75 \times 10^{12}$	$8.00 \times 10^{12}$
6	$6.75 \times 10^{5}$	$3.24 \times 10^{12}$	$8.00 \times 10^{12}$
7	$7.96 \times 10^{5}$	$3.82 \times 10^{12}$	$8.00 \times 10^{12}$
8	$9.40 \times 10^{5}$	$4.51 \times 10^{12}$	$8.00 \times 10^{12}$
9	$1.11 \times 10^{6}$	$5.32 \times 10^{12}$	$8.00 \times 10^{12}$
10	$1.31 \times 10^{6}$	$6.28 \times 10^{12}$	$8.00 \times 10^{12}$
11	$1.54 \times 10^{6}$	$7.41 \times 10^{12}$	$8.00 \times 10^{12}$
12	$1.82 \times 10^{6}$	$8.75 \times 10^{12}$	$8.00 \times 10^{12}$
13	$2.15 \times 10^{6}$	$1.03 \times 10^{13}$	$8.00 \times 10^{12}$
14	$2.54 \times 10^{6}$	$1.22 \times 10^{13}$	$8.00 \times 10^{12}$
15	$2.99 \times 10^{6}$	$1.44 \times 10^{13}$	$8.00 \times 10^{12}$
16	$3.53 \times 10^{6}$	$1.70 \times 10^{13}$	$8.00 \times 10^{12}$
17	$4.17 \times 10^{6}$	$2.00 \times 10^{13}$	$8.00 \times 10^{12}$
18	$4.92 \times 10^{6}$	$2.36 \times 10^{13}$	$8.00 \times 10^{12}$
19	$5.80 \times 10^{6}$	$2.79 \times 10^{13}$	$8.00 \times 10^{12}$
20	$6.85 \times 10^{6}$	$3.29 \times 10^{13}$	$8.00 \times 10^{12}$
21	$8.08 \times 10^{6}$	$3.88 \times 10^{13}$	$8.00 \times 10^{12}$
22	$9.54 \times 10^{6}$	$4.58 \times 10^{13}$	$8.00 \times 10^{12}$
23	$1.13 \times 10^{7}$	$5.40 \times 10^{13}$	$8.00 \times 10^{12}$
24	$1.33 \times 10^{7}$	$6.37 \times 10^{13}$	$8.00 \times 10^{12}$

A

I.

Given $S=4.80\times10^6$ and $N_0=2.50\times10^5$ , find the initial storage required in bytes, in scientific notation.

[2]

II.

Convert this storage requirement to GiB.

[1]

B

The archive has capacity $8.00\times10^{12}$ bytes.

I.

Form an inequality for the number of complete months before the capacity is exceeded.

[2]

II.

Determine the first month in which the capacity is exceeded.

[2]

C

Instead of allowing storage to grow by $18\%$ each month, the archive compresses all images at the end of each month by a fixed factor $q$ , where $0<q<1$ . Thus, for part (c), the modified model is

N_n=N_0(1.18q)^n

The monthly multiplier is therefore $1.18q$ .

I.

Find the greatest value of $q$ that would keep the archive within capacity for $24$ complete months.

[3]

II.

Explain why the inequality sign is not reversed when solving for $q$ .

[1]

Question 34

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

A training app models the time $T_n$ in minutes taken by a student to complete a puzzle on attempt $n$ by

T_n=A+Be^{-kn}

where $A$ is the limiting completion time and $B,k>0$ .

Completion time models decrease toward the limiting time A=6.

A

For one student, $A=6$ , $T_1=22$ and $T_4=14$ .

I.

Show that $e^{-3k}=\dfrac12$ .

[2]

II.

Find $k$ and $B$ .

[3]

B

Use the model found in part (a). If you did not obtain values for $B$ and $k$ , use $B=20.2$ and $k=0.231$ .

I.

Find the first attempt number for which the completion time is less than $8$ minutes.

[2]

II.

Explain why the model never predicts a completion time below $6$ minutes.

[1]

C

A second student has the same limiting time $A=6$ and the same initial excess time as the first student, namely $16\ \text{min}$ . For this student, use the model $T_n=6+B_2e^{-k_2n}$ , where $k_2=1.2k$ and $B_2e^{-k_2}=16\ \text{min}$ .

I.

Find this student's predicted time on attempt $4$ .

[2]

II.

For the model $T_n=A+Be^{-kn}$ , discuss whether increasing $k$ has a larger effect for small $n$ or for large $n$ . Hold $A$ and $B$ fixed and define the effect as the absolute change in $T_n$ when $k$ is increased by $20\%$ .

[2]

Question 35

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Let $x>2$ and

L=\log_3(x-2)+\log_3(x+6)-\log_3 8

A

I.

Express $L$ as a single logarithm.

[2]

II.

Show that the equation $L=2$ can be written as $x^2+4x-84=0$ .

[2]

B

Hence solve $L=2$ , giving your answer in exact form.

[2]

C

Determine the number of solutions of the equation $L=k$ , where $k\in\mathbb{R}$ .

[4]

Question 36

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

The difference, $D$ , between the temperature of an object and the surrounding temperature is modelled by

D(t)=81\left(\frac{2}{3}\right)^t

where $t$ is the time in hours and $D(t)$ is measured in degrees Celsius.

A

I.

Find $D(2)$ .

[2]

II.

Write down the value of $\dfrac{D(t+1)}{D(t)}$ .

[1]

B

I.

Find the exact time when $D(t)=16$ .

[2]

II.

Find the exact solution of the inequality $D(t)<8$ .

[2]

C

Determine the least integer value of $n$ for which $D(n)<1$ .

[3]

Question 37

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A light source emits power $P=3.2\times 10^{26}\ \text{W}$ . At a distance $r$ metres from the source, the intensity $I$ is modelled by

I=\frac{P}{4\pi r^2}

A

I.

Find $r^2$ when $r=4.0\times 10^{11}$ . Give your answer in scientific notation.

[2]

II.

Find the intensity when $r=4.0\times 10^{11}$ .

[2]

B

At what distance is the intensity equal to $\dfrac{5}{\pi}$ ?

[3]

C

Show that multiplying the distance by $10^n$ divides the intensity by $10^{2n}$ .

[3]

Question 38

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The concentration $C$ of a medicine in a patient’s bloodstream is modelled by

C(t)=18a^t

where $t$ is the time in hours after the first dose, $C$ is measured in $\text{mg L}^{-1}$ , and $0<a<1$ . It is known that $C(6)=7.50$ .

A

I.

Find the value of $a$ .

[2]

II.

Find the time taken for the concentration from the first dose to halve.

[2]

B

Determine the threshold-crossing time after the first dose, when the concentration reaches $2.50\ \text{mg L}^{-1}$ .

[3]

C

I.

A second dose is given $8$ hours after the first dose. This adds $10\ \text{mg L}^{-1}$ to the concentration immediately. Find the total concentration immediately after the second dose.

[2]

II.

After the second dose, the total concentration is modelled by the sum of the remaining concentration from each dose. Assume that the concentration from the second dose decays with the same factor $a$ as the first dose. Determine the threshold-crossing time after the first dose, when the total concentration reaches $5.00\ \text{mg L}^{-1}$ . If you did not obtain a value for $a$ , use $a=0.864$ .

[2]

Question 39

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A laboratory investigates the intensity $I$ of light, in lux, after it passes through $d$ millimetres of tinted glass. The data are modelled by

I=Ab^d

where $A>0$ and $0<b<1$ .

d [mm]	I [lux]
0	150.0
1	132.3
2	116.6
3	102.9
4	90.7
5	80.0
6	70.6
7	62.3
8	54.9
9	48.4
10	42.7
11	37.7
12	33.2

A

I.

Use the data to find an exponential regression model for $I$ in the form $I=Ab^d$ .

[2]

II.

Interpret the value of $b$ in the context of the model.

[2]

B

Use your model to estimate the thickness of glass needed to reduce the intensity to $40.0$ lux.

[3]

C

A second piece of glass of the same type is placed after a first piece of thickness $d_1$ . The total thickness is $d_1+d_2$ . Explain why the model predicts that the percentage loss in intensity caused by the second piece depends only on $d_2$ , not on $d_1$ .

[4]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $a>1$ . For real $x$ , define

F(x)=\log_a x+\log_x(a^4)

wherever the expression is defined.

A

I.

State the restrictions on $x$ required for $F(x)$ to be defined.

[1]

II.

Let $u=\log_a x$ . Express $\log_x(a^4)$ in terms of $u$ .

[1]

B

Show that solving $F(x)=k$ is equivalent to solving $u^2-ku+4=0$ , where $u=\log_a x$ .

[3]

C

Hence solve $F(x)=5$ , giving your answers in terms of $a$ .

[2]

D

Determine the values of $k$ for which $F(x)=k$ has exactly two distinct solutions for $x$ .

[3]

Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $x>3$ and $y>1$ . Consider the system

\log_2 x+\log_2 y=5

\log_2(x-3)+\log_2(y-1)=3

A

I.

Show that $xy=32$ .

[2]

II.

Show that $x+3y=27$ .

[2]

B

Hence find all possible ordered pairs $(x,y)$ .

[2]

C

For each ordered pair found in part (b), calculate $\log_2(x+3y)$ and comment on why the value is the same for both pairs.

[3]

Question 42

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A sensor records the brightness $B$ of a lamp after it is switched on. The model used is

B(t)=B_{\text{max}}\left(1-e^{-kt}\right)

where $t$ is the time in seconds, $B_{\text{max}}$ is the limiting brightness and $k>0$ . The graph shows the brightness increasing towards a horizontal asymptote.

Exponential brightness curve approaching a limiting brightness of 80 units.

A

I.

Given that $B_{\text{max}}=80$ and $B(12)=50$ , show that $k=0.0817\ \text{s}^{-1}$ , correct to three significant figures.

[3]

II.

Using the unrounded value of $k$ obtained in part (a)(i), before rounding to three significant figures, find the time at which the brightness first reaches $70$ .

[2]

B

A second lamp follows the model $C(t)=100(1-a^t)$ , where $0<a<1$ . It reaches $60$ units of brightness after $15$ seconds.

I.

Find the exact value of $a$ in the form $m^{1/n}$ , where $m$ is a positive rational number and $n$ is a positive integer.

[2]

II.

Write $C(t)$ in the form $100(1-e^{-\lambda t})$ and find $\lambda$ .

[1]

C

For a general model $D(t)=L(1-e^{-kt})$ , where $L$ is the limiting brightness, define $T_p$ to be the time required to reach $p\%$ of the limiting brightness, where $0<p<100$ .

I.

Show that $T_p=-\dfrac{1}{k}\ln\left(1-\dfrac{p}{100}\right)$ .

[3]

II.

Hence find the exact value of $\dfrac{T_{90}}{T_{50}}$ and interpret this value in context.

[2]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The energy $E$ released by a meteor impact is estimated from its diameter $d$ metres using

E=4.20\times 10^{9}d^3

where $E$ is measured in joules. A logarithmic impact index $I$ is defined by

I=\log\left(\frac{E}{10^{12}}\right)

where $\log$ denotes base $10$ .

Straight-line graph of impact index I against log₁₀ d.

A

I.

Find $E$ when $d=75$ , giving your answer in scientific notation.

[2]

II.

Find the impact index for $d=75$ .

[2]

B

The index can be written in the form $I=A+3\log d$ , where $A$ is a constant.

I.

Show that $A=\log(4.20)-3$ .

[2]

II.

Use this result to determine the diameter of a meteor with impact index $4.00$ .

[2]

C

Two meteors have diameters $d_1$ and $d_2$ . Their impact indices satisfy $I_2-I_1=1.50$ .

I.

Show that $\dfrac{d_2}{d_1}=10^{0.5}$ if the second meteor has the larger index.

[3]

II.

Hence determine the ratio of the released energies $\dfrac{E_2}{E_1}$ , giving your answer in scientific notation.

[2]

III.

Explain why a straight-line graph is obtained when $I$ is plotted against $\log d$ , and state its gradient.

[1]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For a weak acid solution, the hydrogen ion concentration $H$ in moles per litre is related to its pH by

\text{pH}=-\log H

A laboratory dilutes a solution by adding pure water. The concentration after $n$ equal dilution steps is modelled by

H_n=H_0r^n

where $0<r<1$ .

n	H [mol L^-1]	pH
0	3.20×10^-4	3.4949
1	1.60×10^-4	3.7959
2	8.00×10^-5	4.0969
3	4.00×10^-5	4.3979
4	2.00×10^-5	4.6990
5	1.00×10^-5	5.0000
6	5.00×10^-6	5.3010
7	2.50×10^-6	5.6021
8	1.25×10^-6	5.9031
9	6.25×10^-7	6.2041

A

I.

A solution has $H_0=3.20\times10^{-4}$ . Find its pH.

[2]

II.

After $5$ dilution steps, $H_5=1.00\times10^{-5}$ . Find $r$ .

[2]

B

Assume $r=0.500$ for the rest of the question.

I.

Show that the pH after $n$ dilution steps is $3.49+n\log 2$ , correct to three significant figures in the constant term.

[2]

II.

Determine the smallest integer value of $n$ for which the pH exceeds $6.00$ .

[1]

C

A different solution has pH $P_0$ before dilution. Each dilution step multiplies $H$ by the same factor $r$ .

I.

Prove that the increase in pH after one dilution step is $-\log r$ .

[3]

II.

A technician wants each step to increase the pH by exactly $0.250$ . Find the required dilution factor $r$ .

[2]

Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A capacitor discharges through a resistor. The voltage $V$ after time $t$ seconds is modelled by

V(t)=V_0e^{-t/\tau}

where $V_0$ is the initial voltage and $\tau$ is a positive constant called the time constant.

Exponential voltage decay of a discharging capacitor with a 3.00 V threshold line.

A

I.

Given that $V_0=12.0$ and $V(4)=7.20$ , find $\tau$ .

[2]

II.

Using your value of $\tau$ , find the time for the voltage to fall to $1.50$ volts.

[2]

B

Engineers sometimes use base $2$ half-life notation instead of base $e$ notation.

I.

Show that $V(t)=V_0\left(\dfrac12\right)^{t/h}$ , where $h=\tau\ln2$ .

[3]

II.

Find $h$ for this capacitor.

[1]

C

A safety circuit uses a fixed threshold of $3.00$ volts. Define the activation time as the time at which the voltage falls to this threshold. The initial voltage is doubled, while $\tau$ is unchanged.

I.

Find an expression for the increase in activation time caused by doubling $V_0$ .

[3]

II.

Interpret this result in terms of half-life.

[1]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Atmospheric pressure $P$ in kilopascals at height $h$ kilometres above sea level is modelled by

P(h)=101e^{-ch}

where $c>0$ .

Atmospheric pressure decreases exponentially with altitude, with sea-level and summit points marked.

A

I.

At $2.40$ km above sea level, the pressure is $75.0$ kPa. Find $c$ .

[2]

II.

Using this model, find the pressure at $5.50$ km.

[2]

B

Let $R=\dfrac{P(h)}{101\ \text{kPa}}$ be the pressure as a fraction of sea-level pressure.

I.

Show that $h=-\dfrac{\ln R}{c}$ .

[2]

II.

Find the altitude at which the pressure is half the sea-level pressure.

[1]

C

The model may also be written as:

P(h)=101\times10^{-ah}

where $a>0$ .

I.

Express $a$ in terms of $c$ .

[2]

II.

A mountaineer incorrectly uses $a=c$ . Determine the percentage error in the predicted pressure at $6.00$ km, using $c=0.124$ .

[3]

III.

State the mathematical reason for the error.

[1]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A biologist models the metabolic rate $M$ of an animal by a power law

M=kW^p

where $W$ is the animal's mass in kilograms, and $k,p$ are positive constants. Taking logarithms gives a linear relationship.

The exact calibration data are $(W,M)=(4,18)$ and $(W,M)=(64,144)$ . The graph displays the corresponding logarithmic coordinates $(1,2.085)$ and $(3,3.585)$ rounded to three decimal places.

Two calibration points on the log-log M–W relation.

A

For two animals, $(W,M)=(4,18)$ and $(64,144)$ .

I.

Use the two data points to find $p$ exactly.

[3]

II.

Find $k$ exactly.

[2]

B

Use $M=\dfrac{9\sqrt2}{2}W^{3/4}$ .

I.

Find the mass of an animal with metabolic rate $300$ units.

[2]

II.

If mass is multiplied by $16$ , determine the factor by which metabolic rate is multiplied.

[2]

C

Let $X=\\log W$ and $Y=\log M$ , where the logarithm may be taken in any fixed base.

I.

Show that the graph of $Y$ against $X$ is a straight line and state its gradient.

[3]

II.

Justify why the value of the gradient does not depend on the base of the logarithm used.

[2]

Question 48

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $a>0$ , $a\neq 1$ and $x>0$ . Let $y=\log_a x$ .

A

I.

Show that, for $r\neq 0$ , $\log_{a^r}x=\dfrac{y}{r}$ .

[3]

II.

Hence simplify $P=\log_{a^2}x+\log_{a^{-1}}x+\log_{a^{\frac{1}{3}}}x$ in terms of $y$ .

[3]

B

It is given that $P=10$ and $a^2x=32$ .

I.

Find $y$ . If you did not obtain $P=\dfrac{5y}{2}$ , use this expression.

[1]

II.

Find the exact values of $a$ and $x$ .

[3]

III.

Find $\log_x a$ .

[2]

Question 49

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Consider the equation

4^x+4^{-x}=\frac{17}{4}

A

I.

By letting $u=2^x$ , show that the equation may be written as $4u^4-17u^2+4=0$ .

[3]

II.

Solve $4u^4-17u^2+4=0$ for $u>0$ .

[3]

III.

Hence solve the original equation.

[2]

B

Determine all real values of $k$ for which $4^x+4^{-x}=k$ has at least one real solution.

[4]

Question 50

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $x>0$ , $x\neq 1$ . Consider the equation

\log_x 32+\log_8 x=\frac{14}{3}

A

I.

Let $u=\log_2 x$ . Express $\log_x 32$ and $\log_8 x$ in terms of $u$ .

[3]

II.

Show that $u^2-14u+15=0$ .

[2]

B

Hence solve the original equation, giving your answers in exact form.

[4]

C

For $x>1$ , determine the least possible value of $\log_x 32+\log_8x$ .

[3]

Question 51

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $a>0$ , $a\neq 1$ , $x>0$ and $x\neq 1$ . Consider

\log_a x+\log_x a=c

A

I.

Let $u=\log_a x$ . Show that the equation can be written as $u+\dfrac{1}{u}=c$ .

[3]

II.

Find the possible values of $u$ when $c=\dfrac{5}{2}$ .

[2]

B

Given that $a=16$ and $c=\dfrac{5}{2}$ , find all possible values of $x$ .

[2]

C

Assume now that $a>1$ and $x>a$ . Determine all values of $c$ for which the equation has exactly one solution for $x$ .

[5]

Question 52

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $p>0$ , $p\neq 1$ . Consider the expression

Q=p^{2x}-(p+1)p^x+p

A

I.

By letting $t=p^x$ , factorize $Q$ in terms of $t$ .

[2]

II.

Hence solve $Q=0$ for $x$ .

[2]

B

Solve $Q<0$ for $x$ when $p>1$ .

[3]

C

Solve $Q<0$ for $x$ when $0<p<1$ .

[3]

D

Show that $Q<0$ has the same solution set for every permitted value of $p$ .

[2]

Question 53

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider the function

f(x)=2^x+3^{-x},\quad x\in\mathbb{R}

Graph of f(x)=2^x+3^-x with the line y=4.

A

I.

Use a GDC to solve $f(x)=4$ .

[2]

II.

State the interval on which $f(x)<4$ .

[1]

B

Find the minimum value of $f(x)$ and the value of $x$ at which it occurs.

[4]

C

Determine the number of solutions of $2^x+3^{-x}=k$ for different real values of $k$ .

[3]

Question 54

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider the equation

2^{2x}+5\cdot2^{-x}=7

A

I.

Let $u=2^x$ . Show that the equation can be written as $u^3-7u+5=0$ .

[3]

II.

Explain why only positive roots of this cubic are relevant.

[1]

B

Use a GDC to find the positive roots of $u^3-7u+5=0$ .

[2]

C

Hence solve $2^{2x}+5\cdot2^{-x}<7$ .

[3]

Question 55

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

In a chemical experiment, a reaction rate $R$ is modelled by

R=Ae^{-\frac{B}{T}}

where $T$ is the temperature in kelvin, and $A$ and $B$ are positive constants. The rate is $1.80\times10^{-5}$ when $T=290$ , and $6.70\times10^{-4}$ when $T=330$ .

A

I.

Show that $\ln R=\ln A-\frac{B}{T}$ .

[1]

II.

Use the two given rates to show that $B\approx8650$ .

[2]

B

Find the value of $A$ . Give your answer in scientific notation.

[4]

C

Determine the temperature at which the model predicts $R=1.00\times10^{-3}$ . If you did not obtain values for $A$ and $B$ , use $A=1.62\times10^8$ and $B=8650$ .

[4]

Question 56

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The apparent brightness $F$ of a star is measured in watts per square metre. Astronomers define an apparent magnitude $m$ by

m=-2.5\log\left(\frac{F}{F_0}\right)

where $F_0$ is a reference brightness.

Star	Apparent brightness $F$ [W m $^{-2}$ ]	Apparent magnitude $m$
Star A	$3.60\times10^{-6}$	$-5.00$
Star B	$3.60\times10^{-7}$	$-2.50$
Reference	$3.60\times10^{-8}$	$0.00$
Star C	$3.60\times10^{-9}$	$2.50$
Star D	$3.60\times10^{-10}$	$5.00$
Star E	$3.60\times10^{-11}$	$7.50$

A

I.

Given $F_0=3.60\times10^{-8}$ and $F=9.00\times10^{-10}$ , find $m$ .

[2]

II.

Find $F$ when $m=6.00$ , giving your answer in scientific notation.

[2]

B

Two stars have brightnesses $F_1$ and $F_2$ , and magnitudes $m_1$ and $m_2$ .

I.

Show that $m_2-m_1=-2.5\log\left(\dfrac{F_2}{F_1}\right)$ .

[3]

II.

Hence find the brightness ratio $F_2:F_1$ if star $2$ is $3.00$ magnitudes greater than star $1$ .

[2]

C

A cluster contains $N$ identical stars, each with brightness $F$ and magnitude $m$ . The total brightness is $NF$ .

I.

Derive an expression for the magnitude $M_N$ of the cluster in terms of the magnitude $m$ of one star and $N$ .

[3]

II.

If each star has magnitude $8.00$ , find the least integer $N$ for which the cluster has magnitude less than $2.00$ .

[1]

Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

In a chemical kinetics experiment, the rate constant $K$ is modelled by

K=Ae^{-E/(RT)}

where $T$ is the temperature in kelvin and $A,E,R$ are positive constants. Taking natural logarithms gives

\ln K=\ln A-\frac{E}{R}\left(\frac1T\right)

The graph shows experimental values of $\ln K$ plotted against $1/T$ .

Arrhenius plot of ln K against 1/T, with experimental points and a continuous decreasing best-fit line extrapolated to the y-intercept.

A

For one reaction, a line of best fit has equation $y=-6200x+18.4$ , where $y=\ln K$ and $x=\dfrac1T$ .

I.

State the value of $\ln A$ .

[1]

II.

Find $A$ , giving your answer in scientific notation.

[2]

III.

Given $R=8.31$ , find $E$ .

[1]

B

Use the line of best fit from part (a).

I.

Estimate $K$ when $T=350$ K.

[2]

II.

Determine the temperature at which $K=10.0$ .

[2]

C

A student instead plots $\log K$ against $1/T$ , using base $10$ logarithms.

I.

Suppose $\log K$ is plotted against $1/T$ , using common logarithms. Find the gradient and intercept of this new line.

[3]

II.

Explain why the same experimental data still lie approximately on a straight line.

[1]

D

Let $K_1=K(T_1)$ and $K_2=K(T_2)$ , and define $q=\frac{K_2}{K_1}$ . The model predicts that increasing the temperature from $T_1$ to $T_2$ multiplies the rate constant by $q$ .

I.

Show that $\ln q=\dfrac{E}{R}\left(\dfrac1{T_1}-\dfrac1{T_2}\right)$ .

[2]

II.

Using $E/R=6200$ , find $q$ when the temperature increases from $T_1=340\ \text{K}$ to $T_2=360\ \text{K}$ .

[1]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A data scientist studies first digits of numbers that are spread evenly on a logarithmic scale. In this model, the probability that the first digit is $d$ is

P(d)=\log(d+1)-\log d

for $d=1,2,\ldots,9$ , where $\log$ denotes base $10$ .

Bar chart of the first-digit probabilities P(d) for d = 1, 2, ..., 9.

A

I.

Show that $P(d)=\log\left(1+\dfrac1d\right)$ .

[2]

II.

Find $P(1)$ and $P(9)$ correct to three significant figures.

[2]

B

Consider the sum of all nine probabilities.

I.

Prove that $\sum_{d=1}^{9}P(d)=1$ .

[3]

II.

Explain why this supports interpreting $P(d)$ as a probability distribution.

[1]

C

The model is generalised to leading two-digit numbers. For a leading two-digit number $n$ , where $n=10,11,\ldots,99$ , let its probability be

Q(n)=\log(n+1)-\log n

I.

Prove that $\sum_{n=10}^{99}Q(n)=1$ .

[3]

II.

Determine the probability that the leading two-digit number is from $50$ to $59$ inclusive.

[2]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A transparent filter transmits a fraction $T$ of incoming light. Its absorbance $A$ is defined by

A=-\log T

where $0<T\le1$ and $\log$ denotes base $10$ . When filters are stacked, their transmission fractions multiply.

Part	Filter arrangement	Transmission data	Absorbance data	Other given values and condition
(a)(i)	One filter	$T=0.180$	Not given	—
(a)(ii)	One filter	Not given	$A=1.30$	—
(b)(i)	Two filters in series	$T_1$ , $T_2$ ; $T_{\mathrm{total}}=T_1T_2$	$A_1$ , $A_2$	—
(b)(ii)	Three identical filters in series	$T_{\mathrm{total}}=0.200\%=0.00200$	Not given	—
(c)(i)	One filter	Not given	$A=0.37$	—
(c)(ii)	$n$ identical filters in series	Not given	$A=0.37$ each	$I_{\mathrm{in}}=6.50\times10^{-3}\ \mathrm{W\,m^{-2}}$ ; $I_{\mathrm{th}}=1.00\times10^{-6}\ \mathrm{W\,m^{-2}}$ ; activation when $I_{\mathrm{trans}}\le I_{\mathrm{th}}$

A

I.

A filter transmits $18.0\%$ of the incoming light. Find its absorbance.

[2]

II.

Find the transmission fraction of a filter with absorbance $1.30$ .

[2]

B

Two filters have transmission fractions $T_1$ and $T_2$ , and absorbances $A_1$ and $A_2$ .

I.

Prove that the absorbance of the two filters stacked together is $A_1+A_2$ .

[3]

II.

Three identical filters are stacked and transmit $0.200\%$ of the incoming light. Find the absorbance of one filter.

[1]

C

A manufacturer produces filters of absorbance $0.37$ each. The incoming light intensity is $6.50\times10^{-3}\ \text{W m}^{-2}$ . A detector activates when the transmitted intensity is less than or equal to $I_{\mathrm{th}}$ , where $I_{\mathrm{th}}=1.00\times10^{-6}\ \text{W m}^{-2}$ .

I.

Find the transmission fraction of one filter.

[1]

II.

Determine the least number of filters needed for the detector to activate.

[3]

III.

Explain why the inequality sign reverses when isolating $n$ in part (c)(ii).

[2]

Question 60

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $-2<x<5$ , define

f(x)=\log_3(x+2)-\log_3(5-x)

A

I.

State the domain of $f$ .

[1]

II.

Solve $f(x)=0$ .

[1]

B

Show that the inverse function is

f^{-1}(y)=\frac{5\cdot3^y-2}{3^y+1}

[3]

C

Use a GDC to solve $f(x)=2x-1$ . Give all solutions in the domain.

[3]

D

Explain why the answer $x=5.00$ to three significant figures must be interpreted carefully.

[3]