IBMathsDPApplications & Interpretation (AI)HLExam Questions1. Number & AlgebraExponentials & Logs

Exponentials & Logs (DP IB Applications & Interpretation (AI): HL): Exam Questions

3 hours29 questions
Medium
Hard
Very Hard
1a
3 marks

Let l o g subscript 2 space 16 equals l o g subscript 2 a to the power of b comma where a and b are integers and a less than b.

(i) Find the values of a and b.

(ii) Hence, or otherwise, find the value of l o g subscript 2 space 16.

How did you do?

1b
2 marks

Let log space 25 plus log space 4 equals log space c.

(i) Find the value of c.

(ii) Hence, or otherwise, find the value of log space 25 plus log space 4.

How did you do?

1c
2 marks

Let l o g subscript 5 space 500 minus l o g subscript 5 space 4 equals l o g subscript 5 space d.

(i) Find the value of d.

(ii) Hence, or otherwise, find the value of l o g subscript 5 space 500 minus l o g subscript 5 space 4.

How did you do?

2a
2 marks

Let x = \ln 15 and y = \ln 3. Write down the following expressions in terms of x and y.

\ln 5.

How did you do?

2b
2 marks

\ln 45.

How did you do?

2c
3 marks

\ln 135.

How did you do?

3a
2 marks

Let r = \log 2 and s = \log 12. Write down the following expressions in terms of r and s.

\log 24.

How did you do?

3b
3 marks

\log 3

How did you do?

3c
3 marks

log space 72.

How did you do?

4a
2 marks

Simplify the following:

\frac{\left(4 x y^{- 2}\right) \left(- 12 x^{- 4} y^{12}\right)}{6 x^{2} y}

How did you do?

4b
2 marks

open parentheses 2 x to the power of negative 1 end exponent y to the power of negative 2 end exponent close parentheses to the power of negative 3 end exponent open parentheses 4 x squared y cubed close parentheses to the power of 4.

How did you do?

4c
2 marks

\left(\sqrt[2]{\left(9 x^{6} y^{- 2} z^{4}\right)}\right)^{3} \left(3 x y z\right)^{- 2}

How did you do?

5
5 marks

Solve the equation 2 - x \sqrt{3} = \frac{7 x}{\sqrt{3}}, giving your answer in the form \frac{\sqrt{a}}{b} where a and b are positive integers and a is prime. State the values of a and b.

How did you do?

6a
2 marks

Given that l o g subscript a space 8 equals 3.

Find the value of l o g subscript a space 64.

How did you do?

6b
2 marks

Find the value of a.

How did you do?

6c
3 marks

Find the value of log subscript a squared end subscript space 8.

How did you do?

7a
2 marks

Let \log_{b} 3 = x and \log_{b} 16 = y.

Find an expression for \log_{b} 9 in terms of x.

How did you do?

7b
2 marks

Find an expression for log subscript b space 4 in terms of y.

How did you do?

7c
2 marks

Find an expression for log subscript b space 48 in terms of x and y.

How did you do?

8a
2 marks

Show that \frac{\left(4 - 2 \sqrt{x}\right)^{2}}{8 x} can be written as 2 x^{- 1} - 2 x^{- \frac{1}{2}} + \frac{1}{2}.

How did you do?

8b
2 marks

Given that 8 square root of 2 equals 2 to the power of a comma find the value of a.

How did you do?

8c
2 marks

The expression \frac{x\left(2x^{4} - \sqrt{x}\right)}{x^{2}} can be written as 2x^{a} - x^{b}, where a, \; b \in \mathbb{Q}.

Find the value of a and the value of b.

How did you do?

9
5 marks

Solve the equation 16 to the power of x minus 3 open parentheses 4 to the power of x plus 1 end exponent close parentheses equals 28. Write your answer in the form fraction numerator ln space a over denominator ln space b end fraction comma where a and b are integers.

How did you do?

10
2 marks

\sqrt{425} can be written in the form a \sqrt{b}, where a and b are integers and b is as small as possible. Find the values of a and b. Show all of your working.

How did you do?

11a
1 mark

The expression a to the power of 1 fifth end exponent cross times a to the power of 2 over 5 end exponent can be expressed in the form a to the power of p.

Find the value of p.

How did you do?

11b
2 marks

Let a to the power of 1 fifth end exponent cross times a to the power of 2 over 5 end exponent space equals 8.

Find the value of a.

How did you do?

11c
1 mark

The expression b cross times b to the power of negative 3 over 2 end exponent can be written in the form 1 over b to the power of q.

Find the value of q.

How did you do?

11d
2 marks

Let b cross times b to the power of negative 3 over 2 end exponent equals square root of 2

Find the value of b.

How did you do?

12a
2 marks

Let f \left(x\right) = 5 \ln \left(x - 7\right).

Find the values of x for which f \left(x\right) is undefined.

How did you do?

12b
3 marks

The graph of y = f\left(x\right) crosses the x-axis at the point P, with coordinates \left(p, \; 0\right).

Find the value of p.

How did you do?

13a
3 marks

Write the expression 3 \ln 2 - \ln 4 in the form \ln k, where k \in \mathbb{Z}.

How did you do?

13b
2 marks

Hence, or otherwise, solve 3 \ln 2 - \ln 4 = - \ln x.

How did you do?

14
5 marks

Solve 5 space ln space 2 space minus ln space 8 space equals negative ln space x.

How did you do?

15a
2 marks

Let f\left(x\right) = \ln \left(\frac{x}{3} - 1\right).

Find the values of x for which f\left(x\right) is undefined.

How did you do?

15b
3 marks

The graph of y = f\left(x\right) crosses the x-axis at the point A, with coordinates \left(a, \; 0\right).

Find the value of a.

How did you do?

1a
2 marks

Given that 2 to the power of m equals 8 and 2 to the power of n equals 16 comma write down the value of m and of n.

How did you do?

1b
4 marks

Hence or otherwise solve 8 to the power of 2 x plus 1 end exponent equals 16 to the power of 2 x minus 3 end exponent.

How did you do?

2
5 marks

Solve the equation 49 to the power of p plus 4 end exponent equals 35 squared to the power of p for p. Express your answer in terms of ln space 7 and ln space 5.

How did you do?

3
5 marks

Solve the equation 4 to the power of x minus 3 cross times 2 to the power of x plus 2 end exponent equals 64.

How did you do?

4a
2 marks

Simplify the following expressions, giving your answers in the form a x to the power of n where a and n are rational numbers and any fractions are in lowest terms.

8 x to the power of negative 3 end exponent divided by 2 x to the power of negative 1 third end exponent

How did you do?

4b
2 marks

2 over 3 x to the power of 7 over 6 end exponent cross times 5 over 2 x to the power of negative 5 over 3 end exponent

How did you do?

4c
3 marks

open parentheses 3 x to the power of negative 2 over 3 end exponent close parentheses squared divided by 6 x to the power of negative 1 over 6 end exponent

How did you do?

5a
2 marks

Given that y equals 8 over 27 x to the power of 6 comma express each of the following in the form a x to the power of n comma where a and n are constants.

space y to the power of 1 third end exponent

How did you do?

5b
2 marks

y to the power of negative 1 end exponent

How did you do?

5c
3 marks

y to the power of negative 4 over 3 end exponent

How did you do?

6a
3 marks

A block of ice cream in the shape of a cuboid that has been taken out of the freezer and left at room temperature. The rate of decrease of the volume of the block of ice cream can be modelled by the formula

V equals A e to the power of x t end exponent

where A and x are constants and t is the time in minutes after the ice cream was removed from the freezer.

Before it was taken out of the freezer the block of ice cream measured 12 cm by 9 cm by 7 cm. Eight minutes later the block of ice cream measured 8.4 cm by 7.1 cm by 5.8 cm.

By taking logarithms of both sides, show that the rate of decrease of the volume of the block of ice cream can be rewritten as

x t space equals space ln space V space minus space ln space A

How did you do?

6b
4 marks

Find the value of x and A.

How did you do?

7a
5 marks

A zoologist is researching the connection between the mass of a particular breed of mouse, M g, and its heartrate, H bpm.  It is suggested that the relationship can be modelled by the equation

H equals k M to the power of x comma spacewhere x comma space k space element of straight real numbers.

The zoologist takes measurements from two different mice and records the data as follows.

Measurement

M open parentheses g close parentheses

H open parentheses b p m close parentheses

1

17.12

513

2

24.51

401

Find the values of x and k.

How did you do?

7b
3 marks

Use the model to predict

(i) the heartrate for a mouse of 20.2 grams,

(ii) the mass of a mouse that has a heartrate of 420 bpm.

How did you do?

1
6 marks

Solve 27 to the power of 4 x plus 2 end exponent equals 81 to the power of 8 x minus 3 end exponent.

How did you do?

2
6 marks

Solve the equation 216 to the power of k plus 2 end exponent equals 12 to the power of 3 k end exponent for k. Express your answer in terms of ln space 6 and ln space 2.

How did you do?

3
5 marks

Solve the equation 2 cross times 25 to the power of x minus 30 cross times 5 to the power of x minus 1 end exponent space equals space 1.

How did you do?

4a
3 marks

Simplify the following expressions, giving your answers in the form a x to the power of n where a and n are rational numbers and any fractions are in lowest terms.

open parentheses 8 x squared close parentheses to the power of negative 1 third end exponent cross times 1 fourth x to the power of negative 1 third end exponent

How did you do?

4b
3 marks

open parentheses 2 over 9 x to the power of 1 half end exponent cross times 1 over 18 x to the power of negative 3 over 4 end exponent close parentheses to the power of negative 1 fourth end exponent

How did you do?

4c
3 marks

open parentheses 8 x to the power of negative 2 over 3 end exponent close parentheses to the power of blank to the power of 2 over 3 end exponent end exponent over open parentheses 64 x to the power of negative 1 third end exponent close parentheses to the power of blank to the power of 1 third end exponent end exponent

How did you do?

5a
2 marks

Given that y equals 81 over 16 x to the power of negative 12 end exponent comma express each of the following in the form a x to the power of n comma where a and n are constants.

y to the power of 3 over 4 end exponent

How did you do?

5b
2 marks

y to the power of negative 1 half end exponent

How did you do?

5c
3 marks

open parentheses y to the power of 1 half end exponent close parentheses to the power of negative 3 end exponent

How did you do?

6a
4 marks

A company has conducted some product research and believes that their profit (P) made over time (t weeks after opening) can be modelled by the equation

P left parenthesis t right parenthesis equals R left parenthesis t right parenthesis minus C left parenthesis t right parenthesis comma

where

R open parentheses t close parentheses equals C open parentheses t close parentheses to the power of b over 100 t end exponent and C open parentheses t close parentheses equals a to the power of negative b t end exponent plus 1000

The company collects data on their revenue, R, and costs, C, at the end of each week. After exactly one week the company’s costs are $3000 and they make a loss of $1000.

Find the values of a and b.

How did you do?

6b
2 marks

Find the week in which the company first makes a positive profit.

How did you do?

6c
1 mark

Suggest a limitation to the company’s model.

How did you do?

7
6 marks

The rate, R, at which the volume of a cloud created in a science lab increases is related to the air temperature, T, and the air pressure, P, by the equation

R = k T^{x} P^{y}

where x, y and k are real constants.

A scientist takes three measurements and records the results in the table.

Measurement

R (cm³ s⁻¹)

T (°C)

P (kPa)

1

52.23

17.1

101.2

2

45.95

15.9

85.0

3

49.54

14.7

120.0

Find x, y and k.

How did you do?