IB SL TEST - Logarithms and Exponents
Name:
Date:
Paper 1 – No Calculator
Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. Where an answer is incorrect, some marks may be given for a correct
method, provided this is shown by written working. You are therefore advised to show all working.
(1)
Solve the equation log9 81 + log9
1
+ log9 3 = log9 x.
9
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(Total 4 marks)
(2) Write each of the following in its simplest form
ln x
(a) e
(b) e
ln x ln y 

x y
(c) ln e

2
(Total 6 marks)
1
(3)
Simplify this expression, giving the answer as a single logarithm:
log 5 15  log 5 4 2  log 5 2 3
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(Total 4 marks)
(4)
Let p  log a q  log b r  log c

b 
Express log  a 3  in terms of p, q , and r .
 c 
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(Total 4 marks)
(5) [Maximum mark: 7]
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2
(6)
Find the exact value of a :
3
2
log 4 a  log 2 a  log 8 625
5
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(Total 6 marks)
(7)
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(8)
Let f (x) = loga x, x  0.
(a)
Write down the value of
(i)
f (a);
(ii)
f (1);
(iii)
f (a4 ).
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(3)
(a) The diagram below shows part of the graph of f.
On the same diagram, sketch the graph of f−1.
(3)
y
2
1
–2
–1
0
f
1
2
x
–1
–2
(Total 6 marks)
4
CALCULATOR ALLOWED
Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. In particular, solutions found from a graphic display calculator should be
supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of
your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this
is shown by written working. You are therefore advised to show all working.
(9)
Solve the following equations.
(a)
ln (x + 2) = 3.
(b)
102x = 500.
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(Total 6 marks)
(10) A population of ladybugs rapidly multiplies so that the population t days from now is given by
𝐴(𝑡) = 3000𝑒 0.01𝑡 .
a. What is the present ladybug population?
b. How many complete days will it take for the population to reach 3218 ladybugs?
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(Total 6 marks)
5
(11) Initially a tank contains 10 000 litres of liquid. At the time t = 0 minutes a tap is opened, and liquid
then flows out of the tank. The volume of liquid, V litres, which remains in the tank after t minutes
is given by
V = 10 000 (0.933t).
(a)
Find the value of V after 5 minutes.
(1)
(b)
Find how long, to the nearest second, it takes for half of the initial amount of liquid to flow out 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)
(Total 7 marks)
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