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O LevelAdditional MathsCambridgeExam QuestionsAlgebra & FunctionsLogarithmic & Exponential Functions

Logarithmic & Exponential Functions (Cambridge O Level Additional Maths)

Exam Questions

2 hours22 questions
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1
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Given that space log subscript a space p space plus space log subscript a space 5 space – space log subscript a space 4 space equals space log subscript a space 20 space, find the value ofspace p.

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2
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In this question, a comma space b comma space c and d are positive constants.

(i)
It is given that y equals log subscript a open parentheses x plus 3 close parentheses space plus space log subscript a open parentheses 2 x minus 1 close parentheses. Explain why x must be greater than 1 half.

(ii)
Find the exact solution of the equation fraction numerator log subscript a 6 over denominator log subscript a open parentheses y plus 3 close parentheses end fraction equals 2

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3
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The functionspace straight f is defined byspace straight f left parenthesis x right parenthesis space equals space ln left parenthesis 2 x plus 1 right parenthesis space for space x space greater or equal than space 0.

Sketch the graph of y space equals space straight f left parenthesis x right parenthesis and hence sketch the graph of y space equals space straight f to the power of negative 1 end exponent space left parenthesis x right parenthesis on the axes below.

q11-0606-s20-qp-21-additional-maths

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4
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Solve the equation 9 to the power of 5 x end exponent over 27 to the power of x minus 2 end exponent equals 243

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5
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Write 3 space lg space x space plus space 2 minus lg space y as a single logarithm.

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6
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Using the substitution y space equals space 2 to the power of x , or otherwise, solve 2 to the power of 2 x plus 1 end exponent space minus 2 to the power of x plus 1 end exponent space minus 2 to the power of x space plus 1 space equals space 0.

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1
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For variables x and y, plotting ln y against ln x gives a straight-line graph passing through the points left parenthesis 6 comma space 5 right parenthesis and left parenthesis 8 comma space 9 right parenthesis.
Show that y = straight e to the power of p x to the power of q where p and q are integers to be found.

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2
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Variables x and y are such that, when lg y is plotted against x cubed, a straight line graph passing through the points (6,7) and (10,9) is obtained. Find y as a function of x.

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3
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Given that log subscript 2 x plus 2 space log subscript 4 y space equals space 8, find the value of x y.

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4a
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Variables x and y are connected by the relationship y space equals space A x to the power of n, where A and n are constants.

Transform the relationship y space equals space A x to the power of n to straight line form.

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4b
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When ln space y is plotted against ln space x a straight line graph passing through the points (0, 0.5) and (3.2, 1.7) is obtained.

Find the value of n and of A.

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4c
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Find the value of y when x space equals space 11.

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5a
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The population P, in millions, of a country is given by space P equals A cross times b to the power of t, where t is the number of years after January 2000 and A and b are constants. In January 2010 the population was 40 million and had increased to 45 million by January 2013.

Show that b space equals1.04 to 2 decimal places and find A to the nearest integer.

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5b
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Find the population in January 2020, giving your answer to the nearest million.

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5c
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In January of which year will the population be over 100 million for the first time?

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6a
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The number, b, of bacteria in a sample is given by b space equals space P plus Q e to the power of 2 t end exponent , where P and Q are constants and t is time in weeks. Initially there are 500 bacteria which increase to 600 after 1 week.

Find the value of P and of Q.

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6b
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Find the number of bacteria present after 2 weeks.

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6c
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Find the first week in which the number of bacteria is greater than 1 000 000.

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7
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straight f left parenthesis x right parenthesis space equals space 3 plus straight e to the power of x space for space x space element of space straight real numbers
straight g left parenthesis x right parenthesis space equals space 9 x minus 5 space for space x space element of space straight real numbers

Find the exact solution of straight f to the power of negative 1 end exponent space left parenthesis x right parenthesis space equals space straight g apostrophe left parenthesis x right parenthesis .

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1
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Solve the equation 3 to the power of 2 x plus 1 end exponent space plus space 8 left parenthesis 3 to the power of x right parenthesis space – space 3 space equals space 0.

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2
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Solve the equationspace 4 space log subscript y space 2 space plus space log subscript 2 space y space equals space 4 space .

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3
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Write the expression log subscript a 9 plus space left parenthesis log subscript a b right parenthesis left parenthesis log subscript square root of b end subscript 9 a right parenthesis in the form c plus d space log subscript a 9, where c and d are integers.

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4
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Find the exact solution of 3 to the power of 2 x end exponent space minus 3 to the power of x plus 1 end exponent space minus 4 space equals space 0.

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5
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Solve the simultaneous equations

10 to the power of x plus 2 y end exponent space equals space 5,

10 to the power of 3 x plus 4 y end exponent space equals space 50 ,

giving x and y in exact simplified form.

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6
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log subscript a square root of b minus 1 half equals log subscript b a comma space where space a greater than 0 space and space b greater than 0.

Solve this equation for b, giving your answers in terms of a.

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7
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Solve the simultaneous equations.


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

2 log subscript 3 open parentheses x plus 1 close parentheses equals log subscript 3 open parentheses y plus 2 close parentheses

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8a
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It is known that y equals A cross times 10 to the power of b x squared end exponent, where A and b are constants. When lg space y is plotted against x squared, a straight line passing through the points (3.63, 5.25) and (4.83, 6.88) is obtained.

Find the value of A and of b.

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8b
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Using your values of A and b, find the value of y when x space equals space 2,

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8c
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Find the positive value of x when y space equals space 4.

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9a
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log subscript 2 open parentheses y plus 1 close parentheses equals 3 minus 2 space log subscript 2 x

log subscript 2 open parentheses x plus 2 close parentheses equals 2 plus log subscript 2 y


Show that x cubed space plus 6 x squared space minus 32 space equals space 0.

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9b
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Find the roots of x cubed space plus 6 x squared space minus 32 space equals space 0.

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9c
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Give a reason why only one root is a valid solution of the logarithmic equations. Find the value of y corresponding to this root.

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