22 Higher Engineering Mathematics
In Problems 12 to 18 solve the equations:
- log 10 x= 4 [10000]
- lgx= 5 [100000]
- log 3 x=2[9]
- log 4 x=− 2
1
2
[
1
32
]
- lgx=−2[ 0 .01]
- log 8 x=−
4
3
[
1
16
]
- lnx=3[e^3 ]
3.2 Laws of logarithms
There are three laws of logarithms, which apply to any
base:
(i) To multiply two numbers:
log(A×B)=logA+logB
The following may be checked by using a calcu-
lator:
lg10= 1
Also, lg 5+lg 2= 0. 69897 ...
+ 0. 301029 ...= 1
Hence, lg( 5 × 2 )=lg10=lg 5+lg 2
(ii) To divide two numbers:
log
(
A
B
)
=logA−logB
Thefollowing may bechecked using acalculator:
ln
(
5
2
)
=ln2. 5 = 0. 91629 ...
Also, ln5−ln2= 1. 60943 ...− 0. 69314 ...
= 0. 91629 ...
Hence, ln
(
5
2
)
=ln5−ln2
(iii) To raise a number to a power:
logAn=nlogA
Thefollowing may bechecked using acalculator:
lg 5^2 =lg 25= 1. 39794 ...
Also, 2lg 5= 2 × 0. 69897 ...= 1. 39794 ...
Hence, lg 5^2 =2lg5
Here are some worked problems to help understand-
ing of the laws of logarithms.
Problem 10. Write log 4+log 7 as the logarithm
of a single number.
log 4+log 7=log( 7 × 4 )
by the first law of logarithms
=log 28
Problem 11. Write log 16−log 2 as the logari-
thm of a single number.
log16−log 2=log
(
16
2
)
by the second law of logarithms
=log 8
Problem 12. Write 2log 3 as the logarithm of a
single number.
2log3=log 3^2 by the third law of logarithms
=log 9
Problem 13. Write
1
2
log 25 as the logarithm of a
single number.
1
2
log 25=log 25
1
(^2) by the third law of logarithms
=log
√
25 =log 5
Problem 14. Simplify: log64−log128+log32.
64 = 26 , 128 = 27 and 32= 25
Hence, log64−log128+log32
=log2^6 −log2^7 +log2^5