Exponential functions 123
lne^3 x=ln7i.e. 3 x=ln7
from which x=
1
3ln7=0.6486,correct to 4 decimal places.Problem 10. Evaluate the following, each correct
to 5 significant figures: (a)1
2ln4. 7291(b)ln7. 8693
7. 8693(c)3 .17ln24. 07
e−^0.^1762(a)
1
2ln4. 7291 =
1
2( 1. 5537349 ...)=0.77687,
correct to 5 significant figures.(b)ln7. 8693
7. 8693=2. 06296911 ...
7. 8693=0.26215, correct
to 5 significant figures.(c)3 .17ln24. 07
e−^0.^1762=3. 17 ( 3. 18096625 ...)
0. 83845027 ...=12.027,
correct to 5 significant figures.Problem 11. Evaluate the following: (a)lne^2.^5
lg10^0.^5(b)5 e^2.^23 lg2. 23
ln2. 23(correct to 3 decimal places)(a)lne^2.^5
lg10^0.^5=2. 5
0. 5= 5(b)5 e^2.^23 lg2. 23
ln2. 23=5 ( 9. 29986607 ...)( 0. 34830486 ...)
( 0. 80200158 ...)
=20.194, correct to 3 decimal places.Problem 12. Solve the equation 9= 4 e−^3 xto find
x, correct to 4 significant figuresRearranging 9= 4 e−^3 xgives
9
4=e−^3 xTaking the reciprocal of both sides gives
4
9=1
e−^3 x=e^3 xTaking Napierian logarithms of both sides gives
ln(
4
9)
=ln(e^3 x)Since logeeα=α,thenln(
4
9)
= 3 xHence, x=1
3ln(
4
9)
=1
3(− 0. 81093 )=−0.2703,
correct to 4 significant figures.Problem 13. Given 32= 70(
1 −e−t
2)
,
determine the value oft, correct to 3 significant
figuresRearranging 32= 70(
1 −e−t
2)
gives32
70= 1 −e−t
2ande−t(^2) = 1 −
32
70
38
70
Taking the reciprocal of both sides gives
e
t
(^2) =
70
38
Taking Napierian logarithms of both sides gives
lne
t
(^2) =ln
(
70
38
)
i.e.
t
2
=ln
(
70
38
)
from which,t=2ln
(
70
38
)
=1.22, correct to 3 signifi-
cant figures.
Problem 14. Solve the equation
2. 68 =ln
(
4. 87
x
)
to findx
From the definition of a logarithm, since
2. 68 =ln
(
4. 87
x
)
thene^2.^68 =
4. 87
x
Rearranging gives x=
4. 87
e^2.^68
= 4. 87 e−^2.^68
i.e. x=0.3339,
correct to 4 significant figures.
Problem 15. Solve
7
4
=e^3 xcorrect to 4
significant figures