Logarithms

A logarithm is a mathematical function that determines the number needed to raise a certain base number to obtain another number.

Definition

The logarithm to base b of x is denoted as log_b(x) and it is defined as the inverse operation to exponentiation, just as subtraction is the inverse of addition and division is the inverse of multiplication.

In other words, if y = b^x then log_b(y) = x.

For example, since 2^3 = 8, the log base 2 of 8 is 3, or log_2(8) = 3.

Properties of Logarithms

1. Product Rule: log_b(mn) = log_b(m) + log_b(n).
2. Quotient Rule: log_b(m/n) = log_b(m) - log_b(n).
3. Power Rule: log_b(m^n) = n * log_b(m).
4. Change of Base Rule: log_b(a) = log_c(a) / log_c(b), for any positive base c.

Common Logarithms

The most commonly used logarithms are base 10 and base e. Logarithms base 10 are called common logarithms and are usually written as log(x). Logarithms base e are called natural logarithms and are usually written as ln(x), where e is Euler's number, approximately equal to 2.71828.

Applications

Logarithms are used extensively in many areas of mathematics, including algebra, calculus, and complex analysis. They are also used in many practical fields such as computer science, physics, engineering, and economics. For example, they are used in calculations involving rates of change and growth, in solving exponential equations, and in data representation.