O Level Add Math : Expo Log Revisited
1.
2.
2011
Express each of the following as a single logarithm:
(i)
log a 2 + log a 3.
[1]
(ii)
2 log10 x – 3 log10 y.
[3]
Solve the equations
(i)
10p = 0.1,
(ii)
(25k2) 2 = 15,
[1]
1
(iii) t
3.
(i)

1
3

[3]
1
.
2
[2]
Sketch the curve y  2  3x , stating the coordinates of any intersections with
the axes.
(ii)
[3]
The curve y  2  3x intersects the curve y  8x at the point P.
Show that the x-coordinate of P may be written as
1
.
3  log 2 3
4.
(i)
(ii)
[5]
Express each of the following in terms of log 10 x and log 10 y.
(a)
 
log 10  x 
 y
[1]
(b)
log 10 10x 2 y 
[3]
Given that
 
2 log 10  x  = 1 + log 10 10x 2 y  ,
 y
find the value of y correct to 3 decimal places.
KL Ang Dec 2011
[4]
Page 117
O Level Add Math : Indices and Logarithms
5.
6.
7.
(i)
2011
On a single diagram, sketch the curves with the following equations. In each case
state the coordinates of any points of intersection with the axes.
(a)
y = a x, where a is a constant such that a > 1,
[2]
(b)
y = 2b x, where b is a constant such that 0 < b < 1.
[2]
(ii)
The curves in part (i) intersect at the point P. Prove that the x-coordinate of P is
1
.
[5]
log 2 a  log 2 b
(i)
Evaluate log5 15 + log5 20 – log5 12.
(ii)
Given that y = 3 ×102x, show that x = a log10 (by), where the values of the
constants a and b are to be found.
[4]
[3]
Solve the equation


ln 1  x 2  1  2ln x ,
giving your answer correct to 3 significant figures.
8.
(a)
Given that u = log4 x, find, in simplest form in terms of u.
(i)
x,
 
log4 16 ,
x
(iii) log x 8.
(ii)
(b)
Page 118
[4]
[5]
Solve the equation  log3 y   log3  y 2   8 .
2
[4]
O Level Add Math : Expo Log Revisited
9.
10.
2011
Solve the equation
(i)
lg (2x) – lg (x – 3) = 1.
[3]
(ii)
log3 y + 4 log y 3 = 4.
[4]
(i)
Show that the equation
log10  x  5  2  log10 x
may be written as a quadratic equation in x.
(ii)
Hence find the value of x satisfying the equation
log10  x  5  2  log10 x .
11.
[3]
[2]
Solve the simultaneous equations
log3 a = 2 log3 b,
log3 (2a – b) = 1.
[5]
12.
Solve the equation 3x+2 = 3x + 32, giving your answer correct to 3 significant figures.
[4]
13.
Solve the equation
ln (5 – x) = ln 5 – ln x,
giving your answers correct to 3 significant figures.
14.
[4]
It is given that ln (y + 5) – ln y = 2 ln x. Express y in terms of x, in a form not involving
logarithms.
[4]
KL Ang Dec 2011
Page 119
O Level Add Math : Indices and Logarithms
15.
2011
Solve the equation ln (3 – x2) = 2 ln x, giving your answer correct to 3 significant
figures.
[4]
Page 120
O Level Add Math : Expo Log Revisited
2011
Answer keys:
1.
2.
(i)
(ii)
log a 6
2
log10
(i)
(ii)
2
x
x
or lg 3
3
y
y
–1
k = ±3
(iii) t = 8
3.
4.
(i)
See below
5.
(i)
(a) log10 x  log10 y
(b) 1  2log10 x  log10 y
(ii)
y = 0.215
(i)
(ii)
2
(a)
(i) 4 u; (ii) 2 – u; (iii) 3
2u
1
9 or
81
6.
(i)
See below
7.
a= 1,b= 1
3
2
8.
0.582
(b)
9.
10.
(i)
(ii)
3.75
9
11.
(i)
(ii)
x2 + 5x – 100 = 0
≈7.81
12.
≈0.107
b= 3,a= 9
2
4
13.
14.
1.38 or 3.62
y
5
x2  1
15
1.22
KL Ang Dec 2011
Page 121
O Level Add Math : Indices and Logarithms
3.
(i)
y
y = 2(3x)
2
0
x
5.
(i)
y
y = 2b x
2
y = ax
P
1
0
Page 122
x
2011