REVIEW - Exponents and Logs Review Nov Questions

8/2/2019 REVIEW - Exponents and Logs Review Nov Questions

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Exponents and Logs Review-from November questions

1. Let ln a = p, ln b = q. Write the following expressions in terms ofp and q.

(a) ln a3b

(b) ln

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..............................................................................................................................................(Total 6 marks)

2. Solve the equation 43x1 = 1.5625 102.

Working:

Answer:

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(Total 4 marks)

3. A population of bacteria is growing at the rate of 2.3% per minute. How long will it take for the

ba


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size of the population to double? Give your answer to the nearest minute.

Working:

Answer:

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(Total 4 marks)

4. The populationp of bacteria at time t is given byp = 100e0.05t. Note: Do (a) only.

Calculate

(a) the value ofp when t= 0;

(b) the rate of increase of the population when t= 10.

Working:

Answers:

(a) ..

(b) ..(Total 6 marks)


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5. Find the exact value ofx in each of the following equations.

(a) 5x+1 = 625

(b) loga (3x + 5) = 2

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..............................................................................................................................................(Total 6 marks)

6. Let log10P=x , log10Q =y and log10R =z. Express in terms ofx ,y andz.

Working:

Answer:

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(Total 4 marks)

2

310log

QR

P


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7. Letf(x) = 6 sin Tx , andg(x) = 6ex3 , for0exe 2. The graph offis shown on the diagram

below. There is a maximum value at B (0.5, b).

(a) Write down the value ofb.

(b) On the same diagram, sketch the graph ofg.

(c) Solvef(x) =g(x) , 0.5 exe1.5.

(Total 6 marks)

0 1 2

B

x

y

Working:

Answers:

(a) .................................................

(b) .................................................


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8. A machine was purchased for $10000. Its value Vaftertyears is given by V=100000e0.3t. Themachine must be replaced at the end of the year in which its value drops below $1500. Determinein how many years the machine will need to be replaced.

(Total 6 marks)

9. A group of ten leopards is introduced into a game park. After tyears the number of leopards, N, is

modelled byN= 10 e0.4t.

(a) How many leopards are there after 2 years?

(b) How long will it take for the number of leopards to reach 100? Give your answers to anappropriate degree of accuracy.

Give your answers to an appropriate degree of accuracy.

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 4 marks)

10. $1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum

Working:

Answers:

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number of months required for the value of the investment to exceed $3000.

Working:

Answer:

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(Total 6 marks)

11. The mass m kg of a radio-active substance at time thours is given by

m= 4e0.2t.

(a) Write down the initial mass.

(b) The mass is reduced to 1.5 kg. How long does this take?

Working:

Answers:

(a) ..................................................................

(b) ..................................................................

(Total 6 marks)


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12. Solve the equation log9 81 + log9 + log9 3 = log9x.

Working:

Answer:

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(Total 4 marks)

13. Let a = logx, b = logy, and c = logz.

Write log in terms ofa, b and c.

Working:

Answer:

........(Total 6 marks)

9

1

3

2

z

yx


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14. Note: Skip d(ii). Initially a tank contains 10000 litres of liquid. At the time t= 0 minutes a tapis opened, and liquid then flows out of the tank. The volume of liquid, Vlitres, which remains inthe tank aftertminutes is given by

V= 10000 (0.933t).

(a) Find the value ofVafter 5 minutes.(1)

(b) Find how long, to the nearest second, it takes for half of the initial amount of liquid to flowout of the tank.

(3)

(c) The tank is regarded as effectively empty when 95% of the liquid has flowed out.

Show that it takes almost three-quarters of an hour for this to happen.(3)

(d) (i) Find the value of10000Vwhen t=0.001 minutes.

(ii) Hence or otherwise, estimate the initial flow rate of the liquid.Give your answer in litres per minute, correct to two significant figures.

(3)

(Total 10 marks)

15. Letf(x) = logax,x"0.

(a) Write down the value of

(i) f(a);

(ii) f(1);

(iii) f(a4 ).

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(b) The diagram below shows part of the graph off.

On the same diagram, sketch the graph off1.(3)

(Total 6 marks)

2012

2

1

1

2

x

f

y

1

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REVIEW - Exponents and Logs Review Nov Questions