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Explicitly for any real constant Exponential function on the complex plane.

The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right.

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The exponential function satisfies the fundamental multiplicative identity for positive integers n and relates the exponential function to the elementary notion of exponentiation.

The exponential function extends to an entire function on the complex plane.

Euler's formula relates its values at purely imaginary arguments to trigonometric functions.

The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.

One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 This is one of a number of characterizations of the exponential function; others involve series or differential equations.

From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, .