IBMaths: AI HLDPTopic Questions1. Number & Algebra1.2 Exponentials & Logs

Exponentials & Logs (DP IB Maths: AI HL)

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1.2 Exponentials & Logs

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1a
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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.

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1b
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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.

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1c
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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.

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2a
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Let x equals ln space 15 and y equals 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 equals 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
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Solve the equation 2 minus x square root of 3 equals 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 l o g subscript a space 8 equals 3.

Find the value of l o g 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 log subscript b space 3 equals x and space log subscript b 16 equals 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 over denominator 8 x end fraction squaredcan be written as 2 x to the power of negative 1 end exponent minus 2 x to the power of negative 1 half end exponent plus 1 half.

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8b
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Given that 8 square root of 2 equals 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 minus x to the power of b comma where a and b rational numbers. State the value of a and b.

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9
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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.

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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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11a
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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.

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11b
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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.

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11c
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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.

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11d
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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.

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

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

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1b
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Given that point P has coordinates open parentheses p comma space 0 close parentheses, find the value of 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 comma 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 x plus 1 end exponent equals 16 to the power of 2 x 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 minus ln space 4 equals negative 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 squared to the power of p for 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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6a
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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

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6b
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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

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6c
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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

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7a
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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.

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7b
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y to the power of negative 1 end exponent

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7c
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y to the power of negative 4 over 3 end exponent

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8a
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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

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8b
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Find the value of x and A.

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9a
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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.

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9b
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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.

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

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

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1b
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Given that point A has 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 end exponent equals 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 space 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 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.

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5
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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.

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6a
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q6a_1-2_exponentials-_-logs-very-hard-ib-ai-hl-maths

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6b
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q6b_1-2_exponentials-_-logs-very-hard-ib-ai-hl-maths

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6c
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q6c_1-2_exponentials-_-logs-very-hard-ib-ai-hl-maths

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7a
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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

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7b
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y to the power of negative 1 half end exponent

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7c
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open parentheses y to the power of 1 half end exponent close parentheses to the power of negative 3 end exponent

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8a
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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.

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8b
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Find the week in which the company first makes a positive profit.

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8c
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Suggest a limitation to the company’s model.

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9
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The rate, R, of increase of the volume of a cloud created in a science lab is related to the change in air temperature,T , and air pressure, P, by the equation

 R equals k T to the power of x P to the power of y, where x comma space y comma space k space element of space straight real numbers.

A meteorologist takes measurements at three intervals and records the data as follows.

Measurement

  bold italic R bold space stretchy left parenthesis c m cubed s to the power of negative 1 end exponent stretchy right parenthesis  bold italic T bold space stretchy left parenthesis degree C stretchy right parenthesis  bold italic P stretchy left parenthesis kPa stretchy right parenthesis

1

48.75

17.1

101.2

2

46.13

15.9

101.8

3

43.47

14.7

102.5

Find x comma space y spaceand k.

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