Higher Engineering Mathematics

26 NUMBER AND ALGEBRA

log (x−1)+log (x+1)=log (x−1)(x+1),

from the first law of

logarithms

=log (x^2 −1)

2 log (x+2)=log (x+2)^2

=log (x^2 + 4 x+4)

Hence if log (x^2 −1)=log (x^2 + 4 x+4)

then x^2 − 1 =x^2 + 4 x+ 4

i.e. − 1 = 4 x+ 4

i.e. − 5 = 4 x

i.e. x=−^54 or− (^114)
Now try the following exercise.
Exercise 16 Further problems on the laws
of logarithms
In Problems 1 to 8, evaluate the given
expression:

1. log 1010000 [4] 2. log 216 [4]


2. log 5125 [3] 4. log 218 [−3]


3. log 82

[

1

3

]

6. lg 100 [2]


7. log 48

[

1

1

2

]

8. ln e^2 [2]

In Problems 9 to 14 solve the equations:

9. log 10 x= 4 [10000]


10. log 3 x= 2 [9]


11. log 4 x=− 2

1

2

[

±

1

32

]

12. lgx=− 2 [0.01]


13. log 8 x=−

4

3

[

1

16

]

14. lnx=3[e^3 ]

In Problems 15 to 17 write the given expressions

in terms of log 2, log 3 and log 5 to any base:

15. log 60 [2 log 2+log 3+log 5]


16. log

(

16 ×^4

√

5

27

)

[

4 log 2+^14 log 5−3 log 3

]

17. log

(

125 ×^4

√

16

√ 4

813

)

[log 2−3 log 3+3 log 5]

Simplify the expressions given in Problems 18

and 19:

18. log 27−log 9+log 81 [5 log 3]


19. log 64+log 32−log 128 [4 log 2]


20. Evaluate

1

2

log 16−

1

3

log 8

log 4

[

1

2

]

Solve the equations given in Problems 21

and 22:

21. logx^4 −logx^3 =log 5x−log 2x
    [
    x= 2

1

2

]

22. log 2t^3 −logt=log 16+logt
    [t=8]

4.3 Indicial equations

The laws of logarithms may be used to solve cer-

tain equations involving powers—calledindicial

equations. For example, to solve, say, 3x=27, log-

arithms to a base of 10 are taken of both sides,

i.e. log 103 x=log 1027

and xlog 103 =log 10 27,

by the third law of logarithms

Rearranging gives

x=

log 1027

log 103

=

1. 43136 ...

0. 4771 ...

= 3

which may be readily checked

(

Note,

(

log 8

log 2

)

isnotequal to lg

(

8

2

))