1. Definition of Logarithm:
Let a (≠1) and n be positive numbers and x be any real number. Then,
If a^{x} = n,
then the logarithm of number n is written as
log_{ a} n = x
Consider the following examples to become clear of logarithms:

Exponential and Logarithm form

In each of the left cells above, the number ‘n’ whose logarithm is written in the right cell is written in red. Definition of Logarithm:
The logarithm of a number n to a base a is x, where the number n is equal to a raised to the power x.
If a^{x} = n, then
log _{a} n = x
Observe the following results from the above table:

1. the logarithm of 1 to any base is 0
Because if
log_{a} 1 = x , ………………………………(1)
then 1 = a^{x} , and from laws of exponents, we know that
if a^{x} = 1, then x = 0
putting x = 0 in (1), we get
log _{a} 1 = 0

2. the logarithm of any number to itself is 1.
because, if log _{a} a = x,
then a = a^{x} , so x = 1
therefore, log _{a} a = 1 3. logarithm of only positive numbers exists. Note: Logarithm of a number can be negative, but logarithm of a negative number does not exist.
For example,
log _{3} (1/_{3}) = log _{3 }3^{-1}
= -1(log _{3}3)
= -1(1) = -1
But log _{3}(-3) does not exist.
Therefore, in each of the left cells in the above table, the number in red is always positive.Let us now solve a few problems applying the logarithm formulas and concepts learned so far.