If for a positive real number a, a ¹ 1
am = b,
we say that m is the logarithm of b to the base a We write this as 
logab
m ,

“log” being the abbreviation of the word “logarithm” Thus, we have
=
=
=
=
=
=
=
=
3
2
2
3
3
5
0
7
log
8
3,
Since2
8
log
9
2,
Since3
9
log 125
3,
Since5
125
log 1
0,
Since7
1
Laws of Logarithms
In the following discussion, we shall take logarithms to any base a, (a > 0 and a ¹ 1)
First Law: loga
 (mn) = logam + logan
Proof: Suppose that logam = x and logan = y
Then ax= m, ay = n
Hence mn = ax ay = ax+y
It now follows from the definition of logarithms that
loga (mn) = x + y = loga m – loga n
Second Law: loga
 
m
n





 = loga m – logan
Proof: Let logam = x, logan = y
Logarithms
APPENDIX IV
Rationalised 2023-24
146
Chemistry
Then ax = m, ay = n
Hence 
x
x y
ay
m
a
n
a



Therefore
a
a
a
m
log
x
y
log
m
log
n
n










Third Law : loga(mn) = n logam
Proof : As before, if logam = x, then ax = m
Then  


n
n
x
nx
m
a
a


giving loga(mn) = nx = n loga m
Thus according to First Law: “the log of the product of two numbers  is equal to the sum of their logs