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
Example 1: Evaluate:
Example 2: Evaluate:
Third Law:
If are positive rational numbers, then
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
.
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
Eighth Law:
If is a positive real number and
is a positive rational number, then
Ninth Law:
If and
are positive real numbers, then
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.