# Common and Natural Logarithm

Common and natural logarithm have base 10 and base 'e' respectively.There are two bases in logarithm. One base is 10 and other base 'e'. The logarithm with base 'e' is called natural logarithm. The value of

e ≈ 2.71828183. The logarithm with base 10 is called common logarithm.

**In this logarithm the base is 10.**

__Common logarithm :__1) log 10

^{n}= n

2) log 10

^{2}= 2

3) log 10

^{3}= 3

4) log 10

^{-4}= -4

5) log 1000 =log 10

^{3}= 3

6) log 1 = log 10

^{0 }= 0

7) log 0.1 = log 10

^{-1 }= -1

__Natural Logarithm :__It is a logarithm with base 'e'. It is denoted by

**ln (x)**.

If e

^{y}= x

By definition of natural logarithm, here the base is 'e',

So log

_{e}= y OR ln (x) = y

**Rules and properties of common and natural logarithm are same .**

**Product Rule (1st rule) : ln(mn) = ln (m) + ln(n)**

**Proof :**Suppose ln(m) = x and ln(n) = y

e

^{x}= m and e

^{y}= n

∴ mn = e

^{x}. e

^{y}

mn = e

^{x + y}

By definition of natural logarithm

ln(mn) = x + y

ln(mn) = ln(m) + ln(n)

**Example :**Find : ln(7) + log 3

**Solution :**Consider ln 7 + ln 3

ln (7 x 3 ) --------> By 1st rule

= ln 21

∴ ln(21) = ln 7 + ln 3

**Quotient rule (2nd rule): ln(m/n) = ln (m) - ln(n)**

**Proof :**Suppose ln(m) = x and ln(n)= y

e

^{x}= m and e

^{y}= n

∴ m/n = e

^{x}/ e

^{y}

m/n = e

^{x - y}

By definition of natural logarithm

ln(m/n) = x - y

ln(m/n) = ln(m) -ln(n)

**Example :**Find ln (8) - ln(2)

**Solution :**Consider ln 8 - ln 2

ln 8 - ln 2 = ln(8/2) --------> By 2nd rule of natural logarithms

= ln (4)

ln (8) - ln(2) = ln (4)

**Power rule(3rd rule): ln(m)**

^{n}= n ln(m)**Proof :**Suppose ln(m) = x

e

^{x}= m

(e

^{x})

^{n}= (m)

^{n}( taking nth power on both sides)

By definition of natural logarithm

ln(m)

^{n}= n. x

ln (m)

^{n}= n . ln m

**Example :**Find ln (8)

^{2}

**Solution :**ln (8)

^{2}= ln (2

^{3})

^{2}

= ln (2

^{6})

= 6. ln 2 (by 3rd rule )

**4th rule :**ln (1)= 0

**5th rule :**ln (x) is undefined when x≤ 0 so ln(0) is undefined.

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