Logarithm is a mathematical function

A logarithm is a mathematical function that describes the relationship between numbers in terms of exponents. It is the inverse operation of exponentiation. The logarithm of a number to a given base is the power to which the base must be raised to obtain that number.

In mathematical notation, the logarithm of a number "x" to the base "b" is denoted as log_b(x), where "b" is the base and "x" is the number. The logarithm function can be defined for any positive base greater than 1.

The logarithm function has several important properties:

1. Logarithm of a product

$\log_b(xy) = \log_b(x) + \log_b(y)$

The logarithm of a product of two numbers is equal to the sum of the logarithms of the individual numbers.

2. Logarithm of a quotient

$\log_b(x/y) = \log_b(x) - \log_b(y)$

The logarithm of a quotient of two numbers is equal to the difference between the logarithms of the individual numbers.

3. Logarithm of a power

$\log_b(x^a) = a \cdot \log_b(x)$

The logarithm of a number raised to a power is equal to the product of the exponent and the logarithm of the base.

4. Change of base formula

$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$

This formula allows you to convert a logarithm from one base to another.

Logarithms are used in various fields of mathematics, science, and engineering, particularly in situations involving exponential growth or decay, solving equations, analyzing algorithms, and manipulating large numbers. They provide a way to condense large numerical ranges into more manageable scales.