# Logarithm of a Number

## Logarithm of a Number

Exponent to which another number called the base of the logarithm must be raised to obtain a given number.

The integer part of a logarithm is called the characteristic, and its fractional part is called the mantissa.

• A common logarithm is a logarithm to the base 10.
• A natural logarithm – or Napierian logarithm – is a logarithm to the base e.

### Notation

The logarithm of the number a to the base b is written as logb(a), where b is a positive number not equal to 1.

### Properties

• The logarithm of 1 is equal to 0: logb(1) = 0.
• The logarithm of a product xy is equal to the sum of the logarithms of its factors x and y: logb(xy) = logb(x) + logb(y), if x > 0 and y >0.
• The logarithm of a quotient $$\frac{x}{y}$$ is the difference of the logarithms of the dividend and of the divisor: logb $$\frac{x}{y}$$ = logb(x) – logb(y), if x > 0 and y > 0.
• The logarithm of a power xy is equal to the product of the exponent y and the logarithm of x to the base b: logb(xy) = ylogb(x), if x > 0.
• The logarithm of a root $$\sqrt[x]{y}$$ is equal to the logarithm of the number y, whose root is sought, divided by the exponent x: logb($$\sqrt[x]{y}$$) = $$\frac{1}{x}$$ logb (y), if y ≥ 2 and y > 0.

### Example

If the logarithm to the base 10 of 100, written as log10(100), is 2, then: 100 = 10².
Therefore, the mantissa (fractional part) is 0, and the characteristic (integer part) is 2.
The characteristic also gives the order of magnitude of 2000.
In this case, the order of magnitude is 10².
It can be noted that the expression log10(100) = 2 is equivalent to the expression 102 = 100.