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A LevelMaths: PureAQAExam Questions6. Exponentials & Logarithms6.2 Laws of Logarithms

Laws of Logarithms (AQA A Level Maths: Pure)

Exam Questions

3 hours42 questions
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Hard
Very Hard
1
Sme Calculator4 marks

Evaluate

(i)

log subscript 3 space 27

(ii)

log subscript 5 space 625

(iii)

log subscript 2 space 1 fourth

(iv)
log subscript a space a

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2
Sme Calculator4 marks

Write the following in the form a plus b space ln space 2, where a spaceand b are integers to be found.

(i)

3 squared plus ln space 4

(ii)

ln space e to the power of 7 plus ln space 8

(iii)

log space 1000 plus 3 space ln space 16

(iv)
5 open parentheses 3 squared plus ln space 64 close parentheses

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3
Sme Calculator4 marks

Solve the following equations, giving your answer in exact form.

(i)

e to the power of 2 x end exponent equals 5

(ii)
3 e to the power of 1 third x end exponent equals 27

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4
Sme Calculator3 marks

Show that

3 space log subscript a space 4plus 2 space log subscript a space 256 equals 22 log subscript a space 2

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5
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Solve the equation

log subscript x space 16 equals 2

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6
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A square has side length 3 space ln space 4

Show that the perimeter of the square is 24 space ln space 2.

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7
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Write the following in the form a space ln space b, where a spaceand b are integers to be found.

4 space ln space 9 plus 2 space ln space 81 minus 3 space ln space 27

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8
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Solve the equation

7 to the power of 2 x minus 1 end exponent equals 343

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9
Sme Calculator4 marks

Write down the value of

(i)

log subscript 3 space 3

(ii)

ln space e to the power of 6

(iii)

log subscript a space 1

(iv)
log space 1000

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10
Sme Calculator3 marks

Show that

4 space log open parentheses 27 over 16 close parentheses equals 12 space log space 3 minus 16 space log space 2

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11a
Sme Calculator1 mark

Express 42 as a product of its prime factors.

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11b
Sme Calculator2 marks

Hence, or otherwise, show that

ln space 42 equals ln space 7 plus ln space 3 plus ln space 2 space

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12
Sme Calculator2 marks

Sketch the graph of y equals e to the power of x, marking clearly the coordinates of any points where the graph intersects the coordinate axes and stating the equation of any asymptotes.

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1a
Sme Calculator2 marks

Evaluate

log subscript 2 space 4 plus log subscript 3 space 27 minus log subscript 4 space 4

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1b
Sme Calculator3 marks

Evaluate

3 space ln space 2 plus 1 half ln space 81 minus 2 space ln space 3

giving your answer in the form ln space q, where q is an integer to be found.

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2a
Sme Calculator2 marks

Solve the following equations, giving your answers in exact form.

e to the power of x equals 5

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2b
Sme Calculator3 marks

3 e to the power of 2 x end exponent equals 9

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2c
Sme Calculator3 marks

e to the power of 2 x minus 1 end exponent equals 4

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3
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By writing 1 equals log subscript a space a, show that

1 plus 2 space log subscript a space b plus 3 space log subscript a space c equals log subscript a space a b squared c cubed

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4a
Sme Calculator3 marks

Write the following as a single logarithm

2 space log subscript a space 6 plus 3 space log subscript a space 2 minus log subscript a space 4

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4b
Sme Calculator3 marks

Write the following in the form a space ln space b, where a and b are integers to be found.

2 space ln space 3 to the power of 4 plus ln space 3 cubed minus ln space 9

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5
Sme Calculator4 marks

The diagram below shows the length of three sides of a triangle, with each side measured in centimetres.

q5-6-1-laws-of-logarithms-edexcel-a-level-pure-maths-medium

Work out the perimeter of the triangle, giving your answer in the form 2 ln space b, where b is an integer to be found.

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6
Sme Calculator3 marks

Show that

2 space ln space x cubed minus 3 space ln space x squared equals 0

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7a
Sme Calculator2 marks

Solve the equation

5 to the power of 2 x end exponent minus 25 equals 0

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7b
Sme Calculator2 marks

Solve the equation

3 to the power of 2 x minus 1 end exponent equals 4 cubed plus 4 squared plus 1

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8
Sme Calculator5 marks
(i)

On the same axes, sketch the graphs of y equals e to the power of x and y equals e to the power of negative x end exponent.
Label any points of intersection between each graph and the coordinate axes.
Write down the equations of any asymptotes.

(ii)
Write down the equation of the line of reflection between the graphs of y equals e to the power of x and   y equals e to the power of negative x end exponent

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9
Sme Calculator3 marks

Solve the equation

log subscript x open parentheses 5 x minus 6 close parentheses equals 2

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1a
Sme Calculator2 marks

Evaluate

log subscript 2 space 8 squared plus 3 space log subscript 2 space 16 minus 2 space log subscript 2 space 2 to the power of 5

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1b
Sme Calculator3 marks

Evaluate

3 space ln space 2 plus 2 space ln space 5 minus 1 half ln space 10000

giving your answer in the form ln space p.

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2a
Sme Calculator2 marks

Solve the equation

4 to the power of 3 x plus 2 end exponent equals 16 to the power of x plus 6 end exponent

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2b
Sme Calculator3 marks

Solve the equation

4 to the power of 2 x plus 3 end exponent minus 8 equals 92

giving your answer to 3 significant figures.

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3a
Sme Calculator2 marks

Solve the following equations, giving your answers in exact form.

4 e to the power of 3 x minus 2 end exponent equals 12

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3b
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3 e to the power of 2 x end exponent plus 8 equals 14 e to the power of x

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4a
Sme Calculator2 marks

Simplify

2 space ln space 3 to the power of 4 plus ln space 3 cubed minus ln space 9

giving your answer in the form a space ln space b, where a and b are integers to be found.

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4b
Sme Calculator2 marks

Write

2 space log subscript a space x plus 3 space log subscript a open parentheses x plus 1 close parentheses minus log subscript a space 4 open parentheses x plus 2 close parentheses

as a single logarithm.

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5
Sme Calculator5 marks
(i)
On the same axes, sketch the graphs of y equals e to the power of x and y equals ln space x
On each graph, label any points where the graph intersects the coordinate axes.
Write down the equations of any asymptotes for each graph.

(ii)
Write down the line of reflection between the graphs y equals e to the power of x and y equals ln space x.

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6
Sme Calculator3 marks

Solve the equation

5 to the power of 2 x end exponent minus 8 cross times 5 to the power of x plus 12 equals 0

giving your answers in the form log subscript a space end subscript b.

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7
Sme Calculator5 marks

Solve the equation

6 cross times 3 to the power of x minus 1 end exponent equals 6 to the power of 2 x end exponent

giving your answer in the form fraction numerator ln space a over denominator ln space b end fraction, where a and b are integers to be found.

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8
Sme Calculator4 marks

A ship sets sail from a harbour.
After some time, the ship’s position is open parentheses 4 space ln space 3 close parentheses east of the harbour and open parentheses 3 space ln space 3 close parentheses north of the harbour.
Find the direct distance between the ship and the harbour at this time giving your answer in the form open parentheses p space ln space 3 close parentheseskm.

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9
Sme Calculator3 marks

By writing 5 as 5 space ln space e, show that 

5 space ln space 2 plus 5

can be written as 5 space ln space 2 e

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10
Sme Calculator3 marks

Solve the equation

log subscript 3 open parentheses x plus 4 close parentheses equals 4 plus 2 space log subscript 3 space x

giving your answers correct to 3 significant figures.

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11
Sme Calculator3 marks

Solve the equation

2 space log subscript x open parentheses x plus 2 close parentheses equals 3

giving your answer correct to 3 significant figures.

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1a
Sme Calculator2 marks

Evaluate

4 space log subscript 3 space 729 plus 3 space log subscript 2 space 64 squared minus 3 space log space 100 plus ln space e to the power of 6

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1b
Sme Calculator3 marks

Evaluate

1 half space ln space 196 plus 1 third space ln space 125 plus 1 fourth space ln space 81 plus 1 fifth space ln space 32

giving your answer in the form ln space q.

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2
Sme Calculator4 marks

Solve the equation

2 cross times 5 to the power of 2 x plus 1 end exponent plus 21 equals 41 cross times 5 to the power of x

giving your answers in the form  log subscript a space b, where a and b are rational numbers to be found.

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3a
Sme Calculator3 marks

Solve the following equations, giving your answers correct to 3 significant figures.

8 e to the power of 3 x squared minus 1 end exponent equals 12

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3b
Sme Calculator3 marks

e to the power of 3 x end exponent minus 42 equals 2 e to the power of x open parentheses 6 e to the power of x minus 7 close parentheses

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4
Sme Calculator3 marks

Show that

2 space log subscript 3 space x plus log subscript 3 open parentheses x squared minus 1 close parentheses minus 2 space log subscript 3 open parentheses x plus 1 close parentheses identical to log subscript 3 fraction numerator x squared open parentheses x minus 1 close parentheses over denominator open parentheses x plus 1 close parentheses end fraction

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5
Sme Calculator3 marks

Write the following as a single logarithm

2 space log subscript p open parentheses x plus 1 close parentheses plus 3 space log subscript p open parentheses x minus 1 close parentheses minus log subscript p open parentheses x squared minus 1 close parentheses

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6
Sme Calculator5 marks

On the same axes, sketch the graphs of y equals e to the power of 2 x end exponent and y equals begin inline style 1 half end style ln space x

On each graph, label any points where the graph intersects the coordinate axes.

Write down the equations of any asymptotes for each graph.

Explain the significance of the line y equals x.

.

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7
Sme Calculator3 marks

Show that 4 minus ln space 16 can be written in the form 4 space ln open parentheses e over 2 close parentheses

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8
Sme Calculator5 marks

Α triangle is drawn inside a circle such that one side of the triangle is the diameter and all three vertices of the triangle lie on the circumference.

The radius of the circle is open parentheses 3 space ln space 2 close parentheses cm.

The two smallest angles in the triangle are alpha and beta respectively where beta equals 2 alpha.

Find all three sides of the triangle, giving your answers in the form alpha ln space 2.

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9
Sme Calculator3 marks

How many real solutions does the equation have?  Justify your answer.

3 space log subscript x open parentheses x plus 1 close parentheses equals ln space e cubed

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10
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Without using a calculator, show that

log subscript 4 space 8 equals log subscript 9 space 27

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