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# Mathematics-Logarithms

Logarithm in mathematics-The word Logarithm is made up of 2 Greek words.
( 1 ) logos ( means " ratio " ) and
( 2 ) arithmos ( means " number " ).

The concept of ' logarithms ' was introduced by Scottish Mathematician John Napier ( 1550 - 1617 ).

Henry Briggs ( 1556 - 1630 ) introduced the decimal logarithms.

Let us explain via an example;

⇒ 33 = 27
⇒ Apply log on both sides.
⇒ log ( 33 ) = log 27.
⇒ 3 log 3 = log 27.
⇒ 3 = ( log 27 ) / ( log 3 ).
⇒ 3 = log 327.
∴ in general, for any positive real number a, a ≠ 1
ax = m ⇔ log am = x

Definition :: -

Let a be a Positive real number a ( a ≠ 1 ), and x be the unique real number such that ax = m, for a positive real number m, then we say that logarithm of m to the base a is x or x is logarithm of m to the base a, written as

log am = x.

Consider the properties :

1 ) For any positive real number a, a ≠ 1, a0 = 1 ⇔ log a1 = 0
2 ) For any positive real number a, a ≠ 1, a1 = a ⇔ log aa = 1
3 ) For any positive real number a, a ≠ 1, ax = ax ⇔ log aax  = x

Laws of Logarithms ::-

1 ) Product Rule ( First Law )

⇒The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers, with reference to the same base.
i.e., log a( mn ) = log am + log an

⇒Proof :: -
log am = x and log an = y
Then m = ax and n = ay
mn = ax × a⇔ ax + y,

Hence log a( mn ) = x + y = log am + log an

2 ) Quotient Rule ( Second Law )

The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.
i.e., log a( m/n ) = log am - log an .

⇒ Proof ::-
Let  log am = x and log an = y,
Then m = ax and n = ay
m / n = ax / ay ⇔ ax-y
Hence log a( m / n ) = x - y ⇔ log am - log an.

3 ) Power Rule ( Third Law )

The logarithm of the nth power of a number is n times the logarithm of the number.
i.e., log a( mn ) = n × log a

⇒ Proof ::-
Let log am = x, so that m = ax
∴mn = ( ax )n = axn

Hence log a( mn ) = nx = n × log am.

4 ) Base changing Rule ::-

log bm = log am / log a

⇒ Proof ::-
Let x = log bm,
We have bx = m,
Taking logarithm to the base a on both sides, we get log a( bx ) = log a
i.e., x log ab = log am / log ab
i.e, log bm = log am / log ab  