exponentiate

Exponentiate refers to the operation of raising a number to a power. In real-number arithmetic, for a positive base a and a real exponent b, a^b is defined as exp(b ln a). This extends the idea of repeated multiplication, so for an integer n, a^n is the product of n copies of a. Exponentiation also handles negative and fractional exponents by extending the definition, though certain cases require careful treatment or a move to complex numbers.

Notation and basic rules are widely used in mathematics. The common form is a^b, or exp(b ln a). Key properties include a^0 = 1 for a ≠ 0, a^1 = a, (a^m)^n = a^{mn}, and a^{m+n} = a^m a^n. Negative exponents yield reciprocals: a^{-n} = 1/a^n. If the base is negative or the exponent is not an integer, the result may be real or complex depending on the exponent; in real arithmetic it is typically restricted to positive bases for non-integer exponents.

Exponentiation is the inverse operation of the logarithm. For a > 0, log_a(b) denotes the exponent to which a must be raised to obtain b, satisfying a^{log_a(b)} = b. Equivalently, a^x = e^{x ln a} ties exponentiation to the natural exponential function.

Applications of exponentiation appear throughout mathematics and the sciences, including growth and decay models, compound interest, and probability. In calculus, derivatives follow d/dx a^x = a^x ln a, and in algebra, d/da a^b = b a^{b-1} for a > 0. Computationally, very large or very small exponents can cause overflow or underflow, and numerical methods often rely on logarithms or exponentiation by squaring.