An exponential function is a function of the form (f(x)=a^x), where the base (a) is a fixed positive real number, usually with (a\ne1), and the exponent (x) varies. The term also refers specifically to the natural exponential function, (\exp(x)=e^x), whose base is Euler’s number, approximately (2.71828). Exponential functions describe multiplicative change: equal increments in the input produce equal ratios between outputs. This distinguishes them from power functions such as (x^2), whose variable occurs in the base. (openstax.org)
Definition and algebraic structure
For a positive base, exponentiation extends from integer exponents to rational exponents through roots, and then to arbitrary real exponents by continuity. Equivalently, [ a^x=\exp(x\ln a), ] where (\ln) denotes the natural logarithm. Positivity of the base ensures that the function is real-valued for every real input; negative bases do not generally permit this. The case (a=1) gives the constant function (1), conventionally excluded from the nonconstant exponential family. (openstax.org)
The fundamental exponent laws are [ a^{x+y}=a^xa^y,\qquad a^0=1,\qquad a^{-x}=\frac1{a^x}. ] Thus, exponentiation converts addition of inputs into multiplication of outputs. For any fixed increment (h), [ \frac{a^{x+h}}{a^x}=a^h, ] independent of (x). For example, (2^x) doubles whenever (x) increases by one, whereas (2^{-x}) halves. More general expressions (Ca^{kx}) introduce an output scale and an input scale without changing the underlying multiplicative structure. (openstax.org)
Graph and inverse
The domain of (a^x) is all real numbers, and its range is the strictly positive real numbers. Its graph passes through ((0,1)) and never crosses the horizontal axis. It is a continuous function and a strictly monotonic function: increasing when (a>1), decreasing when (0<a<1). For (a>1), its value approaches zero as (x\to-\infty) and grows without bound as (x\to+\infty); for (0<a<1), these directions reverse. (openstax.org)
Strict monotonicity makes the function an injective function. Its inverse function is (\log_a x), defined for positive inputs: [ \log_a(a^x)=x,\qquad a^{\log_a x}=x. ] The exponential and logarithmic graphs are reflections across (y=x). Taking logarithms also converts an exponential relation (y=Ca^x), with (C>0), into the linear relation (\ln y=\ln C+x\ln a). (openstax.org)
Natural exponential and calculus
The natural exponential is distinguished by its derivative: [ \frac{d}{dx}e^x=e^x. ] Consequently, its instantaneous rate of change equals its current value. It is the unique solution of the differential equation [ y'=y,\qquad y(0)=1. ] For a general positive base, [ \frac{d}{dx}a^x=(\ln a)a^x. ] The chain rule gives [ \frac{d}{dx}e^{g(x)}=g'(x)e^{g(x)} ] whenever (g) is differentiable. These identities explain the function’s central role in calculus. (openstax.org)
Differentiation immediately yields the integration formulas [ \int e^{kx},dx=\frac{e^{kx}}k+C,\qquad \int a^x,dx=\frac{a^x}{\ln a}+C, ] for (k\ne0) and (a\ne1). A second differentiation gives ((a^x)''=(\ln a)^2a^x>0), so every nonconstant real exponential is a strictly convex function, including decreasing exponentials. (openstax.org)
Another definition uses the everywhere-convergent power series [ \exp(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!} =1+x+\frac{x^2}{2!}+\cdots, ] where (n!) is a factorial. This is also its Taylor series about zero. In particular, (\exp(x)=1+x+O(x^2)) near zero, providing a local linear approximation. (dlmf.nist.gov)
Growth and decay models
A quantity whose rate of change is proportional to its current amount satisfies [ \frac{dN}{dt}=kN,\qquad N(t)=N_0e^{kt}. ] For (N_0>0), positive (k) produces exponential growth and negative (k) produces exponential decay. The ratio (N'(t)/N(t)=k) is constant, rather than the absolute increment per unit time. Growth has doubling time (\ln2/k); decay has half-life (\ln2/|k|). Both intervals are independent of the starting amount. (openstax.org)
Applications include idealized population growth, radioactive decay, continuously compounded interest, and thermal relaxation. In cooling models, the exponentially changing quantity is the difference between an object’s temperature and the ambient temperature, not necessarily its absolute temperature. Such models depend on assumptions such as a constant proportional rate; changing environmental conditions can invalidate that assumption. (openstax.org)
Complex extension and computation
The same series defines (\exp(z)) for every complex number. It is an analytic function throughout the complex plane and has no zeros. For real (x,y), [ e^{x+iy}=e^x(\cos y+i\sin y). ] Thus, imaginary inputs describe rotation rather than positive real growth. The function has period (2\pi i), so its complex extension is not injective and cannot have a single global inverse. (dlmf.nist.gov)
In numerical analysis, evaluation can use argument reduction before summing the series. For example, write (x=m\ln10+r), choosing an integer (m) so that (|r|\le\frac12\ln10). Then [ e^x=10^me^r. ] The smaller argument makes series evaluation more convenient, while the factor (10^m) restores the required scale. Complex evaluation can similarly separate the real exponential factor from sine and cosine factors. (dlmf.nist.gov)