IB Math 11
Test: Ch. 3, 4, 5 – Exponents & Logs
1. The following diagram shows the graph of
NAME:________________________
y
y = 3–x + 2.
The curve passes through the points (0, a) and (1, b).
(a)
Find the value of
(i)
(0, a)
O
(ii)
(1, b)
a;
x
b.
Diagram not to scale
(b)
Write down the equation of the asymptote to this curve.
(Total 8 marks)
2. Solve log 2 x + log 2 ( x − 2) = 3 , for x > 2.
(Total 7 marks)
3. Let f ( x) = log 3
(a)
x
= log 3 16 − log 3 4 , for x > 0.
2
Show that f ( x) = log 3 2 x .
(2)
(b)
Find the value of f(0.5) and of f(4.5).
(3)
The function f can also be written in the form f(x) =
(c)
ln ax
.
ln b
(i)
Write down the value of a and of b.
(ii)
Hence on graph paper, sketch the graph of f, for –5 ≤ x ≤ 5, –5 ≤ y ≤ 5
y
4
2
x
−4
−2
2
4
−2
−4
(iii) Write down the equation of the asymptote.
(6)
(d)
Write down the value of f–1(0).
(1)
The point A lies on the graph of f. At A, x = 4.5.
(e)
On your diagram, sketch the graph of f–1, noting clearly the image of point A.
(4)
(Total 16 marks)
4. (a)Find log 2 32 .
(1)
(b)
 32 x 
Given that log 2  y  can be written as px + qy, find the value of p and of q.
 8 
(4)
(Total 5 marks)
5. Find the exact solution of the equation 92x = 27(1–x).
(Total 6 marks)
6. Given that log5 x = y, express each of the following in terms of y.
(a)
log 5 x 2
(b)
1
log 5  
 x
(c)
log 25 x
(Total 6 marks)
7. A machine was purchased for $10000. Its value V after t years is given by V =100000e−0.3t. The
machine must be replaced at the end of the year in which its value drops below $1500.
Determine in how many years the machine will need to be replaced.
(Total 6 marks)
1. (a)(i)
y = 3–0 + 2
y=1+2
a=3
(ii)
(M1)
(A1)
(A1) (C3)
y = 3–1 + 2
(M1)
1
+2
3
1
b=2
3
y=
(b)
(A1)
(A1) (C3)
y=2
(A2) (C2)
Note: Award (A1) for y = any constant.
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2.
recognizing log a + log b = log ab (seen anywhere)
e.g. log2(x(x – 2)), x2 – 2x
(A1)
recognizing loga b = x ⇔ ax = b (seen anywhere)
e.g. 23 = 8
(A1)
correct simplification
e.g. x(x – 2) = 23, x2 – 2x – 8
A1
evidence of correct approach to solve
e.g. factorizing, quadratic formula
(M1)
correct working
e.g. (x – 4)(x + 2),
A1
2 ± 36
2
x=4
A2 N3
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3.
(a)
combining 2 terms
(A1)
e.g. log3 8x – log3 4, log3 1 x + log3 4
2
expression which clearly leads to answer given
e.g. log 3
8x
4x
, log 3
3
2
f(x) = log3 2x
(b)
A1
attempt to substitute either value into f
AG
N0 2
(M1)
e.g. log3 1, log3 9
(c)
f(0.5) = 0, f(4.5) = 2
A1A1
(i)
A1A1
a = 2, b = 3
N3 3
N1N1
(ii)
A1A1A1
Note: Award A1 for sketch approximately through
(0.5 ± 0.1, 0 ± 0.1)
A1 for approximately correct shape,
A1 for sketch asymptotic to the y-axis.
(iii) x = 0 (must be an equation)
A1
N3
N1
[6
(d)
(e)
f–1(0) = 0.5
A1
N1 1
A1A1A1A1
Note: Award A1 for sketch approximately through (0 ± 0.1,
0.5 ± 0.1),
A1 for approximately correct shape of the graph
reflected over y = x,
A1 for sketch asymptotic to x-axis,
A1 for point (2 ± 0.1, 4.5 ± 0.1) clearly marked and
on curve.
N4 4
[16
4.
(a)
5
A1
N1
(b)
METHOD 1
 32 x
log 2  y
 8

 = log 2 32 x − log 2 8 y


(A1)
= x log2 32 – y log2 8
(A1)
log2 8 = 3
(A1)
p = 5, q = –3 (accept 5x – 3y)
A1 N3
METHOD 2
32 x
8y
=
(2 5 ) x
(2 3 ) y
(A1)
=
25x
(A1)
23y
= 25x–3y
(A1)
log2 (25x–3y) = 5x – 3y
p = 5, q = –3 (accept 5x – 3y)
A1 N3
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5.
METHOD 1
9 = 32 , 27 = 33
(A1)(A1)
expressing as a power of 3, (32 )2 x = (33 )1− x
(M1)
34 x = 33−3 x
(A1)
4 x = 3 − 3x
(A1)
7x = 3
⇒x=
3
7
(A1) (C6)
METHOD 2
2 x log 9 = (1 − x ) log 27
2x
log 27 
=
=
1 − x log 9 
(M1)(A1)(A1)
3

2
(A1)
(A1)
4 x = 3 − 3x
7x = 3
⇒x=
3
7
(A1) (C6)
Notes:
Candidates may use a graphical method.
Award (M1)(A1)(A1) for a sketch, (A1) for showing the point of
intersection, (A1) for 0.4285…., and (A1) for
3
.
7
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6.
(a)
log5 x2 = 2 log5 x
= 2y
(b)
(M1)
(A1) (C2)
log5 1 = –log5 x
(M1)
x
= –y
(c)
(A1) (C2)
log 5 x
log 5 25
1
= y
2
log25 x =
(M1)
(A1) (C2)
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7.
10 000e− 0.3t = 1500
For taking logarithms
− 0.3t ln e = ln 0.15
t=
(A1)
(M1)
(A1)
ln 0.15
− 0.3
(A1)
= 6.32
(A1)
7 (years)
(A1) (C6)
Note:
Candidates may use a graphical method.
Award (A1) for setting up the correct
equation, (M1)(A1) for a sketch, (A1)
for showing the point of intersection,
(A1) for 6.32, and (A1) for 7.
[6]