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Logarithms

Concept and Laws of Logarithms

Let us suppose we are given 3 numbers: 2, 3 and 9.

Now, we know that 32 = 9

Also, 9=3

The above two expressions are formed by combining 2 and 3, and 2 and 9 respectively to get the third number.

Is there an expression wherein we can combine 3 and 9 to get 2?

3 and 9 can be combined to get 2 as:

Here, ‘log’ is the abbreviated form of a concept called ‘Logarithms’.

The expression can be read as ‘logarithm of 9 to the base 3 is equal to 2’.

In general, if a is any positive real number (except 1), n is any rational number such that , then n is called the logarithm of b to the base a, and is written as.

Thus, if and only if .

is called the exponential form and is called the logarithmic form.

The following are the properties of logarithms.

1. Since a is any positive real number (except 1), an is always a positive real number for every rational number n, i.e., b is always a positive real number.

Thus, logarithms are only defined for positive real numbers.

2. Since

Thus, and

where, a is any positive real number except 1

3. If

Then, and

x = y

Thus,

x = y

4. Logarithms to the base 10 are called common logarithms.

5. If no base is given, the base is always taken as 10.

For example, log 5 = log10 5

Let us consider the following example.

Convert the following into logarithmic form.

(i) 53 = 125

(ii)

There are three standard…

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