Exponential function

An exponential function is a mathematical function of the form: \[ f(x) = a \cdot b^{x} \] where: - \( f(x) \) is the value of the function at \( x \), - \( a \) is a constant that represents the initial value or coefficient, - \( b \) is the base of the exponential function, a positive real number, - \( x \) is the exponent, which can be any real number.

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Codex  0 Created 2026-09-24 Updated 2026-10-14

exp (x) = e^{x}

exp (x + y) = exp (x) exp (y), exp (0) = 1, \frac{d}{d x} exp (x) = exp (x) .

Its values are positive, and it defines a bijection from the real numbers to the positive real numbers. The natural logarithm is its inverse function. These properties connect arithmetic with calculus: adding an amount to the input multiplies the output by a fixed factor, and the local rate of change equals the current value. The exponential series gives one construction; an equivalent characterization is the unique solution of a differential equation

y^{'} = y

y (0) = 1

For a concrete example,

y (t) = 10 e^{0.2 t}

starts at

10

and satisfies

y^{'} = 0.2 y

. The absolute derivative increases with

y

, while the relative rate

y^{'} / y

stays

0.2

. At

t = 5

y = 10 e \approx 27.1828

and

y^{'} \approx 5.43656

. The exponential function therefore models accumulation proportional to what is already present, rather than addition of a fixed amount at every step. A positive constant rate gives exponential growth; a negative one gives exponential decay.

Every positive real base