Logarithms

A logarithm is the inverse operation of exponentiation. For a base b > 0 with b ≠ 1, the logarithm of a positive number x is the exponent y that satisfies b^y = x. The function log_b x is defined for x > 0, and, when b is fixed, maps the positive real numbers to the real numbers. The natural logarithm uses base e and is denoted ln x.

Key properties include the product rule log_b(xy) = log_b x + log_b y, the power rule log_b(x^k) = k log_b x, and the quotient rule log_b(x/y) = log_b x − log_b y. The change of base formula log_b x = log_k x / log_k b allows computation in any base k. Inverse relationships hold between logarithms and exponentials: b^{log_b x} = x and log_b(b^x) = x.

Common values illustrate the concept: log_10 100 = 2, ln e = 1, and log_2 8 = 3. The logarithm is defined for x > 0, and the base b must satisfy b > 0 and b ≠ 1.

Historically, logarithms were introduced in the early 17th century to simplify multiplication and division, with further development by Euler and others leading to the natural logarithm and the constant e. Logarithms have broad uses in mathematics and applied sciences: they turn multiplication into addition, exponentiation into multiplication, and aid in solving exponential growth and decay, data analysis, and various computer algorithms. They are fundamental tools in calculus, statistics, and scientific modeling.