## Logarithm and Exponential Form of a Number

**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.

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