C3
Answers - Worksheet A
EXPONENTIALS AND LOGARITHMS
1
a 20.1
b 0.135
c 13.6
2
a =4
b = eln 3 = 3
c = 2e
3
a x=4
ln 16
=
1
3
d −0.598
e 1.97
d =7
e = ln e−1 = −1 f = −0.5
c x2 = 25
b x = 17
f 0.434
1
x
d
x>0 ∴x=5
4
a x = e15
b ln t = 6
t = e6
d ln 5y = 8
e
5y = e8
y=
5
e8
1
5
1
2
1
2
c x − 4 = e7
x = e7 + 4
x + 3 = e2.5
f 4 − 3x = e11
x = e2.5 − 3
3x = 4 − e11
x = 2e2.5 − 6
x=
(4 − e11)
1
3
b ey = 2
y = ln 2
c 5x = ln 3
x = 15 ln 3
d 4t + 1 = ln 12
e e2x − 3 = 14
f e4 − 5x =
(ln 12 − 1)
1
4
2x − 3 = ln 14
x=
6
x=3
a x = ln 0.7
t=
1
2
x = ln 12 = 2.48
x=
1
15
7
2
4 − 5x = ln
(ln 14 + 3)
b 15x − 7 = e4
a ex = 12
x=
1
c e2
(e4 + 7) = 4.11
1
2
7
3
7
5 − 2x = e 3
x=
1
2
y=
y +3
x = ln 3
4
9
(10 − ee) = −1.72
ln 3 = 0.49
y + 3 = ln
3
2
− 3) = −3.98
11
4
19
5
19
h e3t − 1 = 4
3t − 1 = ln 4
t=
1
3
(ln 4 + 1) = 0.80
2e2x − 11ex + 12 = 0
(2ex − 3)(ex − 4) = 0
ex = 32 , 4
x = ln
11
4
x = e 5 = 44.70
i
ln
, ln 4
 Solomon Press
2x − 5
x
=
1
4
1
2x − 5 = e 4 x
1
(2 − e 4 )x = 5
x=
7
)
11
4
=
ln x =
7
9x
x=
1
3
7
2
f 2 ln x + 3 ln x = 19
(5 − e 3 ) = −2.66
g e4 = 3
9
4
e 10 − 3y = ee
7
2
(4 − ln
1
5
y = 2(ln
d ln (5 − 2x) =
1
3
=
5
1
2 − e4
= 6.98
EXPONENTIALS AND LOGARITHMS
C3
8
3x − 4
x −3
3x − 4
=
x −3
ln
Answers - Worksheet A
= ln 2x
9
2x
e5y − x = 0
⇒
5y = ln x
ln x4 = 7 − y
⇒
4 ln x = 7 − y
3x − 4 = 2x(x − 3)
2x2 − 9x + 4 = 0
sub.
(2x − 1)(x − 4) = 0
x = 12 , 4
∴ x = e 3 = 5.29, y = 0.33
page 2
20y = 7 − y
y = 13
5
for ln (x − 3) to be real, x > 3
∴ x=4
10
a
b
y
y = e−x
c
y
y
y = ex − 1
y = ex
y = 2 + ex
(0, 3)
(0, 2)
y = 2ex
(0, 1)
O
d
x
O
e
y
(0, 1) y = e2x
(0, e−1)
x
O
f
y
x
y
y = ln 3x
y = ex
y = 2 + ln x
(0, 1)
(e−2, 0)
y = ln x
O
(1, 0)
x
O
x
y = −ln x
11
a
y
y = 1 + ln x
y = ln (x + 1)
O
x
b ln (x + 1) = 1 + ln x
ln (x + 1) − ln x = 1
x +1
=
x
x +1
=e
x
ln
1
x + 1 = ex
1 = x(e − 1)
x=
( 13 , 0)
(1, 0)
1
e −1
 Solomon Press
O
y = ln (x − 2)
(3, 0)