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IBMaths: AA HLDPExam Questions1. Number & Algebra1.2 Exponentials & Logs

Exponentials & Logs (DP IB Maths: AA HL)

Exam Questions

3 hours30 questions
Medium
Hard
Very Hard
1a
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Find the value of each of the following, giving your answer as an integer.

ln space e.

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1b
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log subscript 2 space end subscript 16.

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1c
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log space 25 space plus space log space 4.

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1d
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log subscript 5 space 500 space minus space log subscript 5 space 4.

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2a
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Let x space equals space ln space 15 and y space equals space ln space 3. Write down the following expressions in terms of x and y.

ln space 5.

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2b
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ln space 45.

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2c
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ln space 135.

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3a
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Let r equals log space 2 and s space equals space log space 12. Write down the following expressions in terms of r and s.

log space 24.

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3b
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log space 3.

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3c
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log space 72.

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4a
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Simplify the following:

fraction numerator open parentheses 4 x y to the power of negative 2 end exponent close parentheses open parentheses negative 12 x to the power of negative 4 end exponent y to the power of 12 close parentheses over denominator 6 x squared y end fraction

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4b
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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.

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4c
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square root of open parentheses 9 x to the power of 6 y to the power of negative 2 end exponent z to the power of 4 close parentheses end root cubed space open parentheses 3 x y z close parentheses to the power of negative 2 end exponent

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5
Sme Calculator5 marks

Solve the equation 2 minus x square root of 3 space equals space fraction numerator 7 x over denominator square root of 3 end fraction comma giving your answer in the form fraction numerator square root of a over denominator b end fraction where a and b are integers. State the values of a and b.

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6a
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Given that log subscript a space 8 space equals 3.

Find the value of log subscript a space 64.

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6b
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Find the value of a.

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6c
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Find the value of log subscript a squared end subscript space 8.

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7a
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Let l o g subscript b space 3 equals x and log subscript b space 16 space equals space y

Find an expression for log subscript b space 9 in terms of x.

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7b
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Find an expression for log subscript b space 4 in terms of y.

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7c
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Find an expression for log subscript b space 48 in terms of x and y.

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8a
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Show that fraction numerator open parentheses 4 minus 2 square root of x close parentheses squared over denominator 8 x end fraction can be written as 2 x to the power of negative 1 end exponent space minus space 2 x to the power of negative 1 half end exponent plus space 1 half.         

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

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8c
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Show that fraction numerator x open parentheses 2 x to the power of 4 minus square root of x close parentheses over denominator x squared end fraction can be written as 2 x to the power of a space minus space x to the power of b comma where a and b are rational numbers. State the value of a and b.

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9
Sme Calculator5 marks

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

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10
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square root of 425 can be written in the form a square root of b.Find the values of a and b. Show all of your working.

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1a
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Let space f open parentheses x close parentheses space equals 5 ln open parentheses x minus 7 close parentheses.

Find the values of x for which space f open parentheses x close parentheses is undefined.

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1b
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Given that point straight P has coordinates open parentheses p comma space 0 close parentheses, find the value of space p.

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

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

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3a
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Write the expression 3 space ln space 2 minus ln space 4 in the form ln space k, where k element of straight integer numbers.

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3b
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Hence, or otherwise, solve 3 space ln space 2 space minus space ln space 4 space equals space minus ln space x.

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4
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Solve the equation 49 to the power of p plus 4 end exponent equals 35 to the power of 2 p end exponent for space p. Express your answer in terms of ln space 7 and ln space 5.

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5
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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.

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6
Sme Calculator5 marks

It is given that space p equals l o g subscript 124 125 cross times l o g subscript 123 124 cross times l o g subscript 122 123 cross times horizontal ellipsis cross times l o g subscript 5 6.

Given that space p element of straight integer numbers, find the value of space p.

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7
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Solve the equation  log subscript 6 3 plus log subscript 6 2 x equals 2 minus log subscript 6 12.

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8
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Solve the equation log subscript 9 x minus log subscript 9 2 equals 2 plus log subscript 9 5.

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9a
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Find the value of log subscript 4 32 plus log subscript 4 8.

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9b
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Find the value of 64 to the power of log subscript 4 3 end exponent.

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10
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Find the integer values of x andspace y for which x plus y space log subscript 9 5 plus 12 space log subscript 27 15 equals 0

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1a
Sme Calculator2 marks

Let space f open parentheses x close parentheses space equals space ln space open parentheses x over 3 minus 1 close parentheses.

Find the values of x for whichspace f open parentheses x close parentheses is undefined.

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1b
Sme Calculator3 marks

Given that point has A coordinates open parentheses a comma space 0 close parentheses, find the value of a.

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

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3
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Solve 5 space ln space 2 space minus ln space 8 equals negative ln space x.

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4
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Solve the equation 216 to the power of k plus 2 end exponent space equals space 12 to the power of 3 k end exponent for k. Express your answer in terms of ln space 6 and ln space 2.

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5
Sme Calculator5 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 equals 1.

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6
Sme Calculator5 marks

It is given thatspace p space equals space log subscript 242 space 243 cross times log subscript 241 space 242 cross times log subscript 240 space 241 cross times... cross times log subscript 3 space 4.

Given thatspace p element of straight integer numbers comma find the value ofspace p.

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7
Sme Calculator5 marks

Solve the equation log subscript 5 space end subscript x minus log subscript 5 space 4 space equals space 4 plus log subscript 5 space 2.

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8
Sme Calculator7 marks

Solve the equation log subscript 4 space open parentheses 2 minus x close parentheses space equals space log subscript 16 space open parentheses 13 minus 4 x close parentheses.

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9a
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Find the value of log subscript 8 space 2560 minus log subscript 8 space end subscript 5.

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9b
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Find the value of 64 to the power of log subscript 8 space end subscript 5 end exponent.

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10
Sme Calculator6 marks

Find the integer values of x and y for which x plus y space log subscript 343 space 6 minus 10 space log subscript 49 space 42 space equals space 0

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