**GETTING STARTED**

Let us say a^{x }= b, where b > 0, a> 0 and a ≠ 1.

☛ The exponent x to which base a is raised to obtain a number b is called ‘**logarithm of b to the base a**’

Eg. 2^{4} = 16

Hence **log _{2} 16 = 4**

☛ The base of the logarithm, a, has to be a positive real number. Therefore b will also be a positive number.

☛ Many a times, a logarithm will be written without a base. Eg. log (125) = log _{10}(125) = 2.096

It means that the base is 10, and it is called as ‘common log’.

☛ Another expression used frequently is ln (x). It denotes natural log where the base is e (Euler’s number) – 2.71828.

Eg. Ln (7.389) = 2

**Properties of Logarithms:**

(i)log _{x }x =1

Eg. **Log _{24}24** =

**1**

(ii) log _{a}(x. y) = log _{a }x + log _{a}y

Eg. **log _{2}32** = log

_{2}(16 x 2)

= log _{2}16 + log _{2}2 = 4+1 = **5**

(iii) log _{a}(x/y) = log _{a}x – log _{a}y

Eg. **log _{2}(32)** = log

_{2}(64/2)

= log _{2}64 – log _{2}2 = 6 – 1 =**5**

(iv) log _{a}1 = 0

Eg. **log _{10}1** =

**0**

(v) log _{a}(x^{p})= p log _{a }x

Eg.** log _{2}(32)** = 5 log

_{2}(2)

= 5 x 1 = **5**

(vi) log _{a}x = 1/ log _{x}a

Eg. Let us consider log _{32}2

**Log _{32}2** = 1/log

_{2}32 =

**1/5**

(vii) log _{a}x = log _{b}x/log _{b}a = log x/log a

Eg. **Log _{4}64** = log

_{2}64/log

_{2}4

= 6/2 =**3**

Note: The examples have been considered with integers, all for ease of understanding and calculation. Logarithm of a number can be a decimal as well.