The function that equals its own slope is unique
Among real functions, only one sends zero to one and matches its derivative at every point: the exponential, written exp(x) or e^x with e about 2.718. It turns addition into multiplication—and its inverse, the natural logarithm, reverses the trick. Fed complex numbers, it even ties into trigonometry.
The exponential function is the only real function that sends 0 to 1 and whose derivative equals its own value everywhere. It is written e^x or exp(x), the second form preferred when the input is a complicated expression, and the name reflects the input serving as an exponent on the constant e, roughly 2.718. Defining it this way demands proofs of both existence and uniqueness, but its main properties then follow with little effort.
Several definitions that look nothing alike turn out to be equivalent. It is the inverse of the natural logarithm; it is the power series 1 + x + x²/2! + x³/3! and onward, which converges absolutely for every x by the ratio test; and it is the limit of (1 + x/n)^n as the integer n grows without bound. It also obeys exp(x + y) = exp(x) · exp(y), turning sums into products, so the natural logarithm turns products into sums. Other continuous functions with a fixed base satisfy the same equation; the exponential stands out by having slope exactly 1 at zero.
On a graph, e^x always climbs and eventually outruns every power of x, while staying above the horizontal axis and creeping toward it for large negative inputs, so that axis is a horizontal asymptote. The derivative rule means the tangent's slope at each point equals the curve's height there. It is sometimes called the natural exponential function to separate it from exponentiation with other fixed bases and from the wider family of functions whose rate of growth or decay is proportional to their current value.
Feeding it complex numbers links complex multiplication with rotations in the plane and with trigonometry, relations summed up by Euler's formula, e^(iθ) = cos θ + i sin θ. The construction stretches further still, to matrices and to elements of Lie algebras.
Source: Exponential function
