Basic Engineering Mathematics, Fifth Edition

Logarithms 113

15.2 Laws of logarithms

There are three laws of logarithms, which apply to any
base:

(1) To multiply two numbers:

log(A×B)=logA+logB

The followingmay be checked by using a calculator.

lg 10= 1

Also, lg 5+lg 2= 0. 69897 ...+ 0. 301029 ...= 1

Hence, lg( 5 × 2 )=lg 10=lg 5+lg 2

(2) To divide two numbers:

log

(

A

B

)

=logA−logB

The following may be checked using a calculator.

ln

(

5

2

)

=ln2. 5 = 0. 91629 ...

Also, ln5−ln2= 1. 60943 ...− 0. 69314 ...

= 0. 91629 ...

Hence, ln

(

5

2

)

=ln5−ln2

(3) To raise a number to a power:

logAn=nlogA

The following may be checked using a calculator.

lg5^2 =lg25= 1. 39794 ...

Also, 2lg5= 2 × 0. 69897 ...= 1. 39794 ...

Hence, lg5^2 =2lg5

Here are some worked problems to help understanding
of the laws of logarithms.

Problem 10. Write log4+log7 as the logarithm

of a single number

log4+log7=log( 7 × 4 ) by the first law of

logarithms

=log28

Problem 11. Write log16−log2 as the logarithm

of a single number

log16−log2=log

(

16

2

)

by the second law of

logarithms

=log8

Problem 12. Write 2 log 3 as the logarithm of a

single number

2log3=log 3^2 by the third law of logarithms

=log9

Problem 13. Write

1

2

log 25 as the logarithm of a

single number

1

2

log25=log25

1

2 by the third law of logarithms

=log

√

25 =log5

Problem 14. Simplify log 64−log 128 + log 32

64 = 26 , 128 = 27 and 32= 25

Hence, log64−log128+log32

=log2^6 −log2^7 +log2^5

=6log2−7log2+5log2

by the third law of logarithms

=4log2

Problem 15. Write

1

2

log16+

1

3

log27−2log5

as the logarithm of a single number

1

2

log16+

1

3

log27−2log5

=log16

1

(^2) +log27
1
(^3) −log5^2
by the third law of logarithms
=log
√
16 +log^3
√
27 −log25
by the laws of indices
=log4+log3−log25