Basic Engineering Mathematics, Fifth Edition

112 Basic Engineering Mathematics

Here are some worked problems to help understand-

ing of logarithms.

Problem 1. Evaluate log 39

Letx=log 3 9then3x=9 from the definition of

a logarithm,

i.e. 3 x= 32 , from whichx= 2

Hence, log 39 = 2

Problem 2. Evaluate log 1010

Letx=log 10 10 then 10x=10 from the definition

of a logarithm,

i.e. 10 x= 101 , from whichx= 1

Hence, log 1010 = 1 (which may be checked

using a calculator).

Problem 3. Evaluate log 168

Letx=log 16 8then16x=8 from the definition

of a logarithm,

i.e. ( 24 )x= 23 i.e. 2^4 x= 23 from the laws

of indices,

from which, 4 x=3andx=

3

4

Hence, log 168 =

3

4

Problem 4. Evaluate lg0. 001

Letx=lg0. 001 =log 100 .001 then 10x= 0. 001

i.e. 10 x= 10 −^3

from which,x=− 3

Hence, lg 0. 001 =− 3 (which may be checked

using a calculator)

Problem 5. Evaluate lne

Letx=lne=logee then ex=e

i.e. ex=e^1 , from which

x= 1

Hence, lne= 1 (which may be checked

by a calculator)

Problem 6. Evaluate log 3

1

81

Letx=log 3

1

81

then 3x=

1

81

=

1

34

= 3 −^4

from whichx=− 4

Hence, log 3

1

81

=− 4

Problem 7. Solve the equation lgx= 3

If lgx=3 then log 10 x= 3

and x= 103 i.e.x= 1000

Problem 8. Solve the equation log 2 x= 5

If log 2 x=5thenx= 25 = 32

Problem 9. Solve the equation log 5 x=− 2

If log 5 x=−2thenx= 5 −^2 =

1

52

=

1

25

Now try the following Practice Exercise

PracticeExercise 59 Lawsof logarithms

(answers on page 346)

Inproblems1to11,evaluatethegivenexpressions.

1. log 1010000 2. log 2 16 3. log 5125


2. log 2

1

8

5. log 8 26.log 7343


6. lg 100 8. lg 0.01 9. log 48


7. log 273 11. lne^2

In problems 12 to 18, solve the equations.

12. log 10 x= 4 13. lgx= 5


13. log 3 x= 2 15. log 4 x=− 2

1

2

16. lgx=− 2 17. log 8 x=−

4

3

18. lnx= 3