Concept and Laws of Logarithms
Let us suppose we are given 3 numbers: 2, 3 and 9.
Now, we know that 32 = 9
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.
where, a is any positive real number except 1
⇒ x = y
⇒ 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
There are three standard…
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