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# Mathematics-Logarithms

Logarithm in mathematics-The word Logarithm is made up of 2 Greek words.

( 1 )

*( means " ratio " ) and*

**logos**( 2 )

*( means " number " ).*

**arithmos**The concept of

**was introduced by Scottish Mathematician**

*' logarithms '**( 1550 - 1617 ).*

**John Napier***( 1556 - 1630 ) introduced the decimal logarithms.*

**Henry Briggs**Let us explain via an example;

⇒ 3

^{3}= 27

⇒ Apply log on both sides.

⇒ log ( 3

^{3}) = log 27.

⇒ 3 log 3 = log 27.

⇒ 3 = ( log 27 ) / ( log 3 ).

⇒ 3 = log

_{3}27.

∴ in general, for any positive real number a, a ≠ 1

a

^{x}= m ⇔ log

_{a}m = x

Definition :: -

Let a be a Positive real number a ( a ≠ 1 ), and x be the unique real number such that a

^{x}= 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

_{a}m = x.

Consider the properties :

1 ) For any positive real number a, a ≠ 1, a

^{0}= 1 ⇔ log

_{a}1 = 0

2 ) For any positive real number a, a ≠ 1, a

^{1}= a ⇔ log

_{a}a = 1

3 ) For any positive real number a, a ≠ 1, a

^{x}= a

^{x}⇔ log

_{a}a

^{x}= 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

_{a}m + log

_{a}n

⇒Proof :: -

log

_{a}m = x and log

_{a}n = y

Then m = a

^{x}and n = a

^{y}

mn = a

^{x}× a

^{y }⇔ a

^{x + y},

Hence log

_{a}( mn ) = x + y = log

_{a}m + log

_{a}n

**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

_{a}m - log

_{a}n .

⇒ Proof ::-

Let log

_{a}m = x and log

_{a}n = y,

Then m = a

^{x}and n = a

^{y}m / n = a

^{x}/ a

^{y}⇔ a

^{x-y}Hence log

_{a}( m / n ) = x - y ⇔ log

_{a}m - log

_{a}n.

**3 ) Power Rule ( Third Law )**

⇒The logarithm of the n

^{th}power of a number is n times the logarithm of the number.

i.e., log

_{a}( m

^{n}) = n × log

_{a}m

⇒ Proof ::-

Let log

_{a}m = x, so that m = a

^{x}∴m

^{n}= ( a

^{x})

^{n}= a

^{xn}

Hence log

_{a}( m

^{n}) = nx = n × log

_{a}m.

**4 ) Base changing Rule ::-**

⇒ log

_{b}m = log

_{a}m / log

_{a}b

⇒ Proof ::-

Let x = log

_{b}m,

We have b

^{x}= m,

Taking logarithm to the base a on both sides, we get log

_{a}( b

^{x }) = log

_{a}m

i.e., x log

_{a}b = log

_{a}m / log

_{a}b

i.e, log

_{b}m = log

_{a}m / log

_{a}b