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) = log A + log B
The following may 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
= log A − log B
The following may be checked using a calculator.
ln
5
2
= ln 2.5 = 0.91629 ...
Also,
ln 5 − ln 2 = 1.60943 ... − 0.69314 ...
= 0.91629 ...
Hence,
ln
5
2
= ln 5 − ln 2
(3) To raise a number to a power:
logA
n
= n logA
The following may be checked using a calculator.
lg 5 2 = lg25 = 1.39794 ...
Also,
2 lg5 = 2 × 0.69897 ... = 1.39794 ...
Hence,
lg 5 2 = 2 lg5
Here are some worked problems to help understanding
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 − log2 as the logarithm
of a single number
log16 − log 2 = log
16
2
by the second law of
logarithms
= log 8
Problem 12. Write 2 log 3 as the logarithm of a
single number
2 log 3 = 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 log 64 − log 128 + log 32
64 = 2 6 , 128 = 2 7 and 32 = 2 5
Hence,
log 64 − log 128 + log32
= log2
6
− log 2
7
+ log 2
5
= 6 log2 − 7 log2 + 5 log2
by the third law of logarithms
= 4 log2
Problem 15. Write
1
2
log16 +
1
3
log27 − 2 log5
as the logarithm of a single number
1
2
log 16 +
1
3
log 27 − 2 log5
= log 16
1
2 + log 27
1
3 − log 5 2
by the third law of logarithms
= log
√
16 + log
3
√
27 − log 25
by the laws of indices
= log4 + log 3 − log25