## Laws of Logarithms or Logarithm Rules

3. The Four important Laws of Logarithms:
Let a be positive and also not equal to 1.
Let p and q be any two positive numbers i.e., p > 0 and q > 0 and
Let n be any real number.
Then without going into the proofs, let us remember the following four very important laws of logarithms
Law 1. log a (pq) = log a p + log a q
Law 2. log a (p/q ) = log a p – log a q
Law 3. log a (p)n = n log a p
Law 4. a (log a p ) = p

To enable ourselves to easily remember the above four important laws of logarithms, let us put the laws in words:
1. the logarithm to any base of a product of two numbers is equal to the sum of the logarithms of the two numbers to that base.
log (product ) = log (1st factor) + log (2nd factor)

2. the logarithm to any base of a fraction of two numbers is equal to the difference in the logarithm of the numerator and the logarithm of the denominator to that base
log ( fraction ) = log (numerator) – log (denominator)

3. the logarithm to any base of the power of a number is equal to the product of the power and the logarithm of the number to that base.
Now, let us solve some problems based on each of the above four laws of logarithms:
1. Express the following logarithms of numbers as sum of logarithms of their factors.
1. log 2 ( 105 )         2. log 3 (165)
Solution 1:
105 = 3 × 5 × 7
So, log 2 (105) = log 2 (3 × 5 × 7)
log 2 (105) = log 2 3 + log 2 5 + log 2 7

Solution 2:
385 = 5 × 7 × 11
So, log 3 (385) = log 3 (5 × 7 × 11)
log 3 (385) = log 3 5 + log 3 7 + log 3 11

2. Express the following logarithms of fractions as difference of logarithms of numbers
1. log 2 (15/77)         2. log a (pq/rs)

Solution 1:
log 2 (15 /77) = log 2 (15) – log 2 (77)
= log 2 (3 × 5) – log 2 (7 × 11)
= log 2 3 + log 2 5 - log 2 7 - log 2 11

Solution 2:
log a (pq/rs) = log a (pq) - log a (rs)
log a p + log a q - log a r - log a.