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 Similarly, the Second Law says: the log of the ratio of two numbers is the difference of their logs Thus,
the use of these laws converts a problem of multiplication/division into a problem of addition/subtraction,
which are far easier to perform than multiplication/division That is why logarithms are so useful in
all numerical computations