## How do Logarithms Work?

### Logarithms

“The Logarithm of a given number to a given base is the index of the power to which the base must be raised in order to equal the given number.”

If a > 0 and ≠ 1 then logarithm of a positive number N is defined as the index x of that power of ‘a’ which equals N i.e.,

It is also known as fundamental logarithmic identity.

Its domain is (0, ∞) and range is R. a is called the base of the logarithmic function.

When base is ‘e’ then the logarithmic function is called **natural** or **Napierian** **logarithmic function** and when base is 10, then it is called common logarithmic function.

### Characteristic and mantissa

- The integral part of a logarithm is called the characteristic and the fractional part is called mantissa.

- The mantissa part of log of a number is always kept positive.
- If the characteristics of log
_{10}N be n, then the number of digits in N is (n+1). - If the characteristics of log
_{10}N be (– n) then there exists (n – 1) number of zeros after decimal part of N.

### Properties of logarithms

Let *m* and *n* be arbitrary positive numbers such that a > 0, a ≠ 1, b > 0, b ≠ 1 then

### Logarithmic inequalities

(10) log _{p}*a* > log _{p}*b* ⇒ *a* ≥ *b* if base *p* is positive and >1 or *a* ≤ *b* if base *p* is positive and < 1 *i.e.,* 0 < *p* < 1.

In other words, if base is greater than 1 then inequality remains same and if base is positive but less than 1 then the sign of inequality is reversed.