Definition: If is a positive number, other than and is a rational number such that , then we say that logarithm of to the base is . It is written as .

i.e.

Some examples for you to grasp this concept a little more:

*Note: For any positive real number a, we know that . Hence and and*

Fundamental Laws of Logarithms:

*First Law: *

If are positive rational numbers, then i.e. the logarithms of the product of two numbers is equal to the sum of their logarithms.

We can use this law and extrapolate it to multiple situations like:

- If are positive numbers, then
- If are positive numbers, then
- If are real numbers such that , then

Example 1: Prove

LHS

= RHS

Example 2: Prove

LHS

RHS

*Second Law: *

If are positive rational numbers, then i.e. the logarithms of the ratio of two numbers is equal to the difference of their logarithms.

The above law can also be stated as follows: If are real numbers such that , then

Example 1: Evaluate:

Example 2: Evaluate:

*Third Law:*

If are positive rational numbers, then

Example 1:

Example 2:

*Fourth law:*

i.e. the log of to any base is always .

*Fifth law:*

i.e. the log of any positive number to the same base is always .

Example 1: Evaluate:

Example 2: Evaluate:

*Sixth law (Base change formula):*

If is a positive rational number and are positive real numbers such that and , then .

In the above formula if b = m, then we get

Example 1: Evaluate:

Since we get

Example 2: Evaluate:

*Seventh Law:*

If is a positive real number and is a positive rational number, then

Example 1:

Example 2:

*Eighth Law:*

If is a positive real number and is a positive rational number, then

Example 1:

Example 2:

*Ninth Law:*

If and are positive real numbers, then

Example 1:

Example 2:

As you would have noticed, we have listed all the key concepts of logarithms (as related to Class 9 students). We have also presented a couple of example / applications of the laws. However, we would recommend the student to solved all the problems as stated in Exercise 8(a). More you practice, the better.