EXPONENTIAL AND LOGARITHMIC FUNCTIONS
6)
7)
8)
9)
10)
11)
12)
13)
14)
Evaluate e2.7 correct to two decimal places.
Sketch the curve y = e-x
Differentiate 4ex
Differentiate xex
x
Find the derivative of x
e
x 9
Differentiate (1 + e )
Differentiate e5x
Find the derivative of e3x + 2x .
Find the exact gradient of the tangent to the curve y = ex at the point where x = 3.
Find the equation of the tangent to the curve y = e2x at the point (1,e2).
Find the equation of the normal to the curve y = 2e4x + 1 at the point where x = 0.
Find the stationary point on the curve y = (x +1) ex and determine its nature.
Find the indefinite integral (primitive function) of e7x.
Find  ( e 2 x  e x ) dx
15)
Evaluate
16)
Find the exact value of
17)
Find the exact area bounded by the curve y = 2e2x, the x-axis and the lines x = 1 and
x=3
Find the area enclosed between the curve y = e3x+1, the x-axis and the lines x = 0 and
x = 2 correct to 3 significant figures..
Find the exact volume of the solid formed when the curve y = e2x is rotated about
the x-axis from x = 1 to x = 3.
x
3e
Use Simpson’s rule with five function values to find an approximation to 
dx to
1 x
two decimal places.
Evaluate log10 56.3 to one decimal place.
Evaluate ln 6.89 to two significant figures.
Evaluate log2 32.
Evaluate log5 125.
Evaluate log36 6.
Evaluate log7 7.
Evaluate log9 1.
1
Evaluate log2
16
Sketch the curve y = ln x on a number plane.
Solve logx 16 = 2
Solve log3 y = 4
Simplify loga p + logaq - logar
1)
2)
3)
4)
5)
18)
19)
20)
21)
22)
23)
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25)
26)
27)
28)
29)
30)
31)
32)

1
0
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e3x dx to two decimal places.

5
1
e5x dx
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33)
34)
35)
36)
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38)
39)
40)
41)
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52)
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54)
55)
Simplify 2logn3 + logny
Evaluate log8 4 + log8 16.
Show that log a 8 + 2log a 3 - log a 6 = log a 12.
Given log5 3 = 0.68 and log5 4 = 0.86, find
(a) log5 12
(b) log5 0.75
(c) log5 27
(d) log5 15
If logp x = n and logp y = t, find logp xy2 in terms of n and t.
Evaluate log4 7 to one decimal place.
Solve 2x = 5 to three significant figures.
Solve 52x+1 = 3 to two decimal places.
Solve 2ex = 13 to two significant figures.
Consider the equation x = 3e2t.
(a) Find x when t = 2, to one decimal place.
(b) Find t when x = 100, to one decimal place.
d2x
(c) Show that
 4x .
dt 2
Differentiate loge (x+1)
Differentiate (ln x + x2)4
Differentiate xlnx.
2x  3
Differentiate loge
x 1
Differentiate loge (ex + 1)
Find the equation of the tangent to the curve y = loge x at the point (4,loge 4).
Show that the function f(x) = ln (2x + 3) has no stationary points.
1
Find the indefinite integral (primitive function) of
x3
4
dx
Evaluate 
to two decimal places.
1 2x  1
3
x
Find the exact value of  2 dx
2 x 1
1
(a) Find the exact area bounded by the curve y = , the x-axis and the lines x = 2
x
and x = 5.
(b) This area is rotated about the x-axis. Find the volume of the solid formed.
The curve y = ln x is rotated about the y-axis from y = 0 to y = 2. Find the exact
volume of the solid of revolution.
(a) Find the approximate area bounded by the curve y = ln x, the x-axis and the lines
x = 1 and x = 3 by using Simpson’s rule with three function values (to two decimal
places).
(b) By considering an area involving the y-axis, find the exact area in part (a).
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ANSWERS
1)
2)
14.88
y
4
3
2
1
x
-3
-2
-1
-1
1
2
3
-2
-3
-4
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
4ex
xex + ex = ex (x + 1)
e x  xe x 1  x
 x
e2x
e
9ex (1 + ex )8
5e5x
3e3x + 2
e3
2e2 x - y - e2 = 0
x + 8y - 24 = 0
(-2,-e2) minimum
1 7x
e C
7
1 2x
e  e x  C
2
6.36
1 5 20
e ( e  1)
5
e2 ( e4  1) units2
365 units2
 4 8
e ( e  1) units3
4
8.04
1.8
1.9
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23)
24)
25)
26)
27)
28)
29)
5
3
1
2
1
0
-4
y
2
1
x
-2
-1
1
2
3
-1
-2
30)
31)
32)
33)
34)
35)
36)
37)
38)
39)
40)
41)
42)
x=4
y = 81
pq
loga
r
logn 9y
2
LHS = log a 8 + 2log a 3 - log a 6
= log a 8 + log a 32 - log a 6
8  32
= log a
6
= log a 12
= RHS
So log a 8 + 2log a 3 - log a 6 = log a 12.
(a) 1.54 (b) -0.18 (c) 2.04 (d) 1.68
n + 2t
1.4
x = 2.32
x = -0.16
x = 1.9
(a) x = 163.8 (b) t = 1.8
(c) x = 3e2t
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43)
44)
45)
46)
47)
48)
49)
50)
51)
52)
53)
54)
55)
dx
 6e 2 t
dt
d 2x
 12 e 2 t
2
dt
= 4(3e2t)
= 4x
1
x 1
1
4(  2 x )(ln x  x 2 ) 3
x
1 + ln x
2
1
5


2 x  3 x  1 ( 2 x  3)( x  1)
ex
ex  1
x - 4y + 4loge 4 - 4 = 0
2
f 1 (x) =
 0 for any x.
2x  3
ln (x - 3) + C
0.97
1  8
ln   C
2  3
3
 5
(a) ln  units2 (b)
units3
 2
10
 4
( e  1) units3
2
(a) 1.29 (b) 3ln3 - 2 units2
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