International ASMathsEdexcelPure 2Exam QuestionsExponentials & LogarithmsLaws of Logarithms

Laws of Logarithms (Edexcel International AS Maths: Pure 2): Exam Questions

Exam code: XMA01

2 hours34 questions
Easy
Medium
Hard
Very Hard
1
Sme Calculator4 marks

Evaluate

(i) log subscript 3 27

(ii) log subscript 5 625

(iii) log subscript 2 1 fourth

(iv) log subscript a a

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2
Sme Calculator3 marks

Show that 3 space log subscript a 4 plus 2 space log subscript a 256 equals 22 space log subscript a 2.

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3
Sme Calculator2 marks

Solve the equation

log subscript x 16 equals 2

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4
Sme Calculator2 marks

Solve the equation

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

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5
Sme Calculator3 marks

Show that

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

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6
Sme Calculator3 marks

Write down the value of:

(i) 3 cubed

(ii) 4 to the power of negative 2 end exponent

(iii) 9 to the power of 0.5 end exponent

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7
Sme Calculator3 marks

Write down the value of a in the following statements:

(i) 3 to the power of a equals 27

(ii) a to the power of 1 third end exponent equals 5

(iii) 4 a squared equals 64

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8
Sme Calculator3 marks

Write down the value of a in the following statements:

(i) log subscript 3 a equals 4

(ii) log subscript a 216 equals 3

(iii) log subscript 2 128 equals a

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9
Sme Calculator1 mark

Solve the equation

2 to the power of x equals 16

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10a
Sme Calculator2 marks

Solve the equation x squared minus 12 x plus 27 equals 0.

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10b
Sme Calculator3 marks

Hence, or otherwise, solve the equation left parenthesis 3 to the power of x right parenthesis squared minus 12 left parenthesis 3 to the power of x right parenthesis plus 27 equals 0.

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11
Sme Calculator2 marks

Solve the equation

2 log subscript 3 9 equals 5 x minus 6

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12
Sme Calculator2 marks

Use a calculator to find the value of

(i) 5 log subscript 3 7

(ii) 2 log subscript 2 3 plus 3 log subscript 3 2

giving your answers to four significant figures.

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1
Sme Calculator3 marks

By writing 1 equals log subscript a a ,show that

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

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2a
Sme Calculator2 marks

Solve the equation

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

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2b
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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3
Sme Calculator3 marks

Solve the equation

log subscript x left parenthesis 5 x minus 6 right parenthesis equals 2.

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4
Sme Calculator4 marks

On the same axes, sketch the graphs of y equals 2 to the power of x and y equals 3 to the power of xcommalabelling any points where the graphs cross the coordinate axes and writing down the equation of any asymptotes.

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5a
Sme Calculator3 marks

Using a calculator, find to 3 significant figures

(i) log subscript 2 5 plus log subscript 5 2

(ii) log 25 minus ln space 2

(iii) log 200 plus log subscript 5 50 minus log 20 plus ln 10

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5b
Sme Calculator2 marks

Solve 3 log subscript 2 4 plus 3 x equals 5 log subscript 6 216.

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6
Sme Calculator3 marks

Solve 2 to the power of 2 x end exponent minus 24 left parenthesis 2 to the power of x right parenthesis plus 128 equals 0.

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1a
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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1b
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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2
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 comma

giving your answers in the form log subscript a b

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3
Sme Calculator3 marks

Solve the equation

log subscript 3 left parenthesis x plus 4 right parenthesis equals 4 plus 2 log subscript 3 x

giving your answers correct to 3 significant figures.

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4
Sme Calculator3 marks

Solve the equation

2 log subscript x left parenthesis x plus 2 right parenthesis equals 3

giving your answer correct to 3 significant figures.

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5a
Sme Calculator4 marks

On the same axes, sketch the graphs of y equals 4 to the power of xand y equals 5 to the power of x.

Label any points of intersection with the coordinate axes.

Write down the equations of any asymptotes.

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5b
Sme Calculator1 mark

Write down an equation for the graph that is a reflection of y equals 4 to the power of x in the y-axis.

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6a
Sme Calculator1 mark

Find the value of log 1000 plus log 10   000

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6b
Sme Calculator1 mark

Write down the value of a in the statement 6 to the power of log subscript 6 a end exponent equals 36

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6c
Sme Calculator2 marks

Evaluate fraction numerator 2 log subscript 4 64 plus 3 log subscript 2 8 minus log subscript 5 5 over denominator log 100 end fraction.

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7a
Sme Calculator2 marks

Solve 2 log 1000 equals x log subscript 16 4

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7b
Sme Calculator2 marks

Solve 3 log subscript 4 x equals log subscript 4 x plus 3 log subscript 5 25.

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8
Sme Calculator3 marks

Solve 2 left parenthesis 2 to the power of 2 x end exponent right parenthesis plus 4 equals 9 left parenthesis 2 to the power of x right parenthesis

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1
Sme Calculator8 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 comma

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

How did you do?

2
Sme Calculator3 marks

Show that

2 log subscript 3 x plus log subscript 3 left parenthesis x squared minus 1 right parenthesis minus 2 log subscript 3 left parenthesis x plus 1 right parenthesis identical to log subscript 3 open parentheses fraction numerator x squared left parenthesis x minus 1 right parenthesis over denominator x plus 1 end fraction close parentheses.

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3
Sme Calculator3 marks

Write the following as a single logarithm:

2 log subscript p left parenthesis x plus 1 right parenthesis plus 3 log subscript p left parenthesis x minus 1 right parenthesis minus log subscript p left parenthesis x squared minus 1 right parenthesis.

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4
Sme Calculator3 marks

Without using a calculator, show that

log subscript 4 8 equals log subscript 9 27.

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5a
Sme Calculator4 marks

On the same axes, sketch the graphs of y equals 0.3 to the power of x and y equals 0.5 to the power of x.
Label any points of intersection with the coordinate axes.
Write down the equations of any asymptotes.

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5b
Sme Calculator1 mark

Write down an equation for the graph that is a reflection of y equals 0.5 to the power of x in the y-axis.

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6
Sme Calculator4 marks

Solve 3 to the power of 2 left parenthesis x plus 1 right parenthesis end exponent plus 3 equals 28 left parenthesis 3 to the power of x right parenthesis.

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7
Sme Calculator4 marks

Find two values of x for which log left parenthesis x squared right parenthesis equals left parenthesis log x right parenthesis squared is true.

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8
Sme Calculator7 marks

Solve the equation log subscript 4 left parenthesis 2 minus x right parenthesis equals log subscript 16 left parenthesis 13 minus 4 x right parenthesis.

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