Exp
In mathematics, the exponential function, commonly denoted as exp(x) or e^x, is a fundamental function defined as the unique real-valued function that maps zero to one and whose derivative equals itself at every point, representing exponential growth or decay proportional to the function's current value.[1] It is specifically given by exp(x) = e^x, where e is Euler's number, the base of the natural logarithm, approximately equal to 2.718281828459.[2] The function exp(x) has a domain of all real numbers and a range of all positive real numbers, serving as the inverse of the natural logarithm function ln(x).[3] One of the defining properties of exp(x) is its series expansion: exp(x) = ∑_{n=0}^∞ x^n / n!, which converges for all real x and provides a way to compute its values without relying on the base e directly.[4] This power series highlights its smoothness and analytic nature, making it central to calculus, differential equations, and complex analysis.[5] Additionally, exp(x) satisfies key functional equations, such as exp(x + y) = exp(x) · exp(y) and exp(0) = 1, which underscore its role in modeling multiplicative processes.[6] The exponential function is ubiquitous in science and engineering, describing phenomena like population growth, radioactive decay, compound interest, and electrical circuits, where rates of change are proportional to the quantity involved.[7] In more advanced contexts, it extends to the complex plane as exp(z) = e^z for complex z, forming the basis for Euler's formula e^{iθ} = cos(θ) + i sin(θ), which links exponential and trigonometric functions.[8] Its importance stems from these properties, positioning exp(x) as a cornerstone of mathematical modeling across disciplines.[9]Definitions
Differential equation approach
One historical motivation for the exponential function arises from the modeling of continuous compounding interest, as investigated by Jacob Bernoulli in 1683. Bernoulli considered the growth of an investment where interest is compounded infinitely often, leading to a scenario where the instantaneous rate of change of the principal is proportional to the principal itself. This proportionality results in the differential equation \frac{dP}{dt} = r P(t), where P(t) is the value at time t and r is the interest rate.[10][11] To solve this equation, apply separation of variables: \frac{dP}{P} = r \, dt. Integrating both sides yields \ln |P| = r t + C, so P(t) = K e^{r t} for some constant K > 0. With initial condition P(0) = P_0, it follows that K = P_0, giving P(t) = P_0 e^{r t}. For the standard exponential function, set r = 1 and P_0 = 1, yielding the initial value problem f'(x) = f(x) with f(0) = 1, whose solution is f(x) = e^x.[12][11] The uniqueness of this solution follows from the Picard–Lindelöf theorem, which guarantees a unique local solution to the initial value problem y' = g(x, y), y(x_0) = y_0, when g is continuous and Lipschitz continuous in y. Here, g(x, y) = y satisfies these conditions globally, ensuring the solution e^x is unique on \mathbb{R}.[13] The base e is defined as the value f(1), representing the unique factor by which the solution grows over the interval [0, 1]. This characterization aligns with the property that the derivative equals the function itself, distinguishing e \approx 2.71828 as the base for which the limit defining the derivative at zero equals 1.[14]Power series definition
The exponential function \exp(x) can be defined for all real numbers x by its power series expansion around zero: \exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}. This representation arises as the Taylor series of \exp(x), where the coefficients are determined by repeated differentiation: assuming \exp(x) is infinitely differentiable with \exp^{(n)}(x) = \exp(x) for all n \geq 0 and \exp(0) = 1, the Taylor coefficients at zero are \exp^{(n)}(0)/n! = 1/n!.[15] The series converges for every real x, as established by the ratio test applied to the absolute values of consecutive terms: \lim_{n \to \infty} \left| \frac{x^{n+1}/(n+1)!}{x^n/n!} \right| = \lim_{n \to \infty} \frac{|x|}{n+1} = 0 < 1. This infinite radius of convergence reflects the rapid growth of the factorial denominator n!, which outpaces any polynomial growth in the numerator.[15][16] Evaluating the series at x = 1 yields the base of the natural logarithm, e = \exp(1) = \sum_{n=0}^{\infty} \frac{1}{n!}, a fundamental constant approximately equal to 2.71828 whose partial sums provide increasingly accurate approximations due to the series' convergence properties.[15] Term-by-term differentiation of the power series reproduces the original series, confirming that \exp(x) satisfies the differential equation y' = y with initial condition y(0) = 1. Since the power series converges absolutely for every complex number z, it defines the analytic continuation of \exp(x) to the entire complex plane, rendering \exp(z) an entire function with no singularities.[15]Limit of powers definition
The exponential function \exp(x) for real x is defined as the limit \exp(x) = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n, where n is a positive integer.[17][18] To establish the existence of this limit, consider the case x > 0. The binomial theorem expands \left(1 + \frac{x}{n}\right)^n = \sum_{k=0}^n \binom{n}{k} \left(\frac{x}{n}\right)^k, revealing that the sequence a_n = \left(1 + \frac{x}{n}\right)^n is increasing because each term in the expansion for a_{n+1} exceeds the corresponding term in a_n.[18][19] Furthermore, the sequence is bounded above, as the expansion shows a_n < \sum_{k=0}^\infty \frac{x^k}{k!} < \infty by comparing to a geometric series after bounding the coefficients.[18][19] Thus, by the monotone convergence theorem, the limit exists and is finite.[18] For x = 0, the limit is trivially 1, and for x < 0, the limit exists by considering the reciprocal.[18] In the specific case x = 1, the constant e is defined as e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.71828. [17] Numerical approximations illustrate the convergence: for n = 10, the value is about 2.59374; for n = 100, about 2.70481; for n = 1000, about 2.71692; and for n = 10^6, about 2.718280.[17] This limit definition extends to powers via manipulation: \exp(xy) = \lim_{n \to \infty} \left(1 + \frac{xy}{n}\right)^n = \lim_{n \to \infty} \left[ \left(1 + \frac{x}{n}\right)^n \right]^y = [\exp(x)]^y for real y.[17] Historically, this formulation ties to Jacob Bernoulli's 1683 investigation of compound interest, where continuous compounding led to the limit \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n as the effective growth factor.[10]Inverse logarithm definition
The natural logarithm function, denoted \ln x, is defined for x > 0 as the definite integral \ln x = \int_1^x \frac{1}{t} \, dt. This integral representation establishes \ln x as a strictly increasing and continuous function, with derivative \frac{d}{dx} \ln x = \frac{1}{x} > 0 for x > 0, ensuring it is bijective from (0, \infty) onto (-\infty, \infty).[20][21][22] Consequently, \ln x possesses a unique inverse function, called the exponential function and denoted \exp y or e^y, defined on all real numbers y \in \mathbb{R} with range (0, \infty), satisfying \exp(\ln x) = x for all x > 0 and \ln(\exp y) = y for all real y.[23][2] This inverse relation guarantees that \exp y > 0 for every real y, as the range of \ln x covers all reals while its domain is the positive reals.[2][24] The uniqueness of \exp y as the inverse stems directly from the monotonicity and continuity of \ln x, which ensure a one-to-one correspondence without ambiguity in the inversion.[23] This definition highlights the mutual exclusivity of the exponential and natural logarithm in real analysis, where the exponential maps reals to positives and the logarithm reverses this mapping. For a general base b > 0, b \neq 1, the exponential function b^x is expressed via the change of base as b^x = \exp(x \ln b), allowing extension from the natural base while preserving the core inverse properties.[25] As a consequence of this inverse relationship, the exponential function satisfies the differential equation y' = y with initial condition y(0) = 1.[23]Fundamental Properties
Algebraic properties
The exponential function satisfies the fundamental functional equation \exp(x + y) = \exp(x) \exp(y) for all real numbers x and y. This identity establishes the exponential as a group homomorphism from the additive group of real numbers to the multiplicative group of positive real numbers. A proof follows from the power series representation \exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!}, which converges for all complex z. Multiplying the series for \exp(x) and \exp(y) yields the Cauchy product \sum_{n=0}^\infty \sum_{k=0}^n \frac{x^{n-k} y^k}{(n-k)! k!}, and applying the binomial theorem gives \sum_{n=0}^\infty \frac{(x+y)^n}{n!} = \exp(x+y).[26] Setting x = [0](/page/0) in the functional equation implies \exp([0](/page/0)) = \exp([0](/page/0)) \exp([0](/page/0)), so \exp([0](/page/0)) (\exp([0](/page/0)) - 1) = [0](/page/0). Since the power series at z = [0](/page/0) evaluates to 1, it follows that \exp([0](/page/0)) = 1 > [0](/page/0). Substituting y = -x yields \exp(x) \exp(-x) = \exp([0](/page/0)) = 1, so \exp(-x) = 1 / \exp(x). The exponential function is strictly positive for all real x, as the power series terms are all nonnegative for x \geq [0](/page/0) (hence \exp(x) \geq 1 > [0](/page/0)), \exp([0](/page/0)) = 1 > [0](/page/0), and \exp(x) = 1 / \exp(-x) > [0](/page/0) for x < [0](/page/0).[15][27] The functional equation extends to integer multiples: \exp(n x) = [\exp(x)]^n for any integer n. For positive integers, this holds by induction, as \exp((n+1)x) = \exp(n x + x) = \exp(n x) \exp(x) = [\exp(x)]^{n+1}, with the base case \exp(x) = [\exp(x)]^1. The case n = 0 follows from \exp(0) = 1 = [\exp(x)]^0. For negative integers, use \exp(-n x) = 1 / \exp(n x) = [\exp(x)]^{-n}. This identity generalizes to rational exponents: for r = p/q with integers p, q (q > 0), let y = \exp(x/q), then y^q = \exp(x), so y = [\exp(x)]^{1/q} (the unique positive real q-th root, as \exp > 0), and \exp(r x) = \exp(p \cdot (x/q)) = [ \exp(x/q) ]^p = [\exp(x)]^r.[27][26] Beyond these identities and the power series expansion, the exponential function has no closed-form expression in terms of algebraic functions; it is transcendental, meaning it cannot be expressed as a finite combination of polynomials and rational functions using the field operations of addition, subtraction, multiplication, and division.[28]Analytic properties
The exponential function \exp(x), often denoted e^x, satisfies the differential equation \frac{d}{dx} \exp(x) = \exp(x), making it equal to its own derivative.[29] This property follows from the functional equation \exp(x + y) = \exp(x) \exp(y), which underpins the limit definition of the derivative.[18] Consequently, all higher-order derivatives of \exp(x) are also \exp(x), as repeated differentiation yields the same result each time.[29] The indefinite integral of the exponential function is given by \int \exp(x) \, dx = \exp(x) + C, where C is the constant of integration, mirroring its differentiation behavior.[29] For real arguments, \exp(x) is strictly increasing because its derivative \exp(x) > 0 for all x, ensuring that \exp(x_1) < \exp(x_2) whenever x_1 < x_2.[18] This monotonicity implies that \exp(x) is bijective from \mathbb{R} to (0, \infty), with its inverse being the natural logarithm function \ln(y).[30] In the complex plane, the exponential function extends to \exp(z) for z \in \mathbb{C}, which is an entire function—holomorphic everywhere with no singularities.[31]Growth and behavior
The exponential function e^x exhibits distinct asymptotic behaviors at the extremes of its domain. As x \to \infty, e^x \to \infty, demonstrating unbounded growth. Conversely, as x \to -\infty, e^x \to 0, approaching the x-axis without reaching it.[32][33] The graph of y = e^x is a smooth curve that passes through the point (0, 1) and lies entirely above the x-axis for all real x. It is strictly increasing everywhere, as confirmed by its analytic derivative being positive, and convex, meaning the curve bends upward and any line segment connecting two points on the graph lies above the curve. At the origin, the graph is tangent to the line y = x + 1.[34][35][36] Compared to polynomial functions, e^x grows faster than any polynomial of finite degree as x \to \infty; for example, \lim_{x \to \infty} \frac{e^x}{x^n} = \infty for any fixed n. This rapid growth underscores its superpolynomial nature.[37] While aperiodic on the real numbers—lacking any real period that repeats its values—the complex exponential function e^z is periodic with period $2\pi i, leading to oscillatory behavior in the imaginary direction.[38]Generalizations and Extensions
Exponential functions with other bases
The exponential function with an arbitrary base b > 0, b \neq 1, is defined for all real exponents x byb^x = \exp(x \ln b),
where \exp denotes the natural exponential function and \ln is the natural logarithm. This definition leverages the natural exponential as the foundational form while extending its structure to other positive bases excluding 1, ensuring the function remains positive and well-defined over the reals.[39] Key algebraic properties hold under this definition, mirroring those of the natural exponential. For instance, the addition formula
b^{x+y} = b^x b^y
applies for all real x and y, derived from the property \exp(u + v) = \exp(u) \exp(v) with u = x \ln b and v = y \ln b. Similarly, b^{xy} = (b^x)^y and b^{-x} = 1 / b^x follow analogously. The base change formula relates exponentials across different bases using logarithms. Specifically, for bases a > 0, a \neq 1, and b > 0, b \neq 1,
b^x = a^{x \log_a b},
where \log_a b = \ln b / \ln a. To prove this equivalence, take the natural logarithm of both sides of the proposed identity: \ln(b^x) = x \ln b. Substituting the right side yields \ln(a^{x \log_a b}) = x \log_a b \cdot \ln a = x (\ln b / \ln a) \ln a = x \ln b, confirming the equality since the natural exponential is one-to-one.[25] Special cases illustrate practical uses. When b = 10, the function $10^x is the inverse of the common logarithm \log_{10}, employed in decimal-based scientific calculations and engineering contexts.[41] For b = 2, $2^x models binary processes, such as population growth via binary fission in microbiology, where each cycle doubles the quantity.[42] These functions inherit strong analytic properties from their construction. Since \exp and \ln are continuous and infinitely differentiable on the reals, the composition b^x is continuous for all real x and differentiable with derivative
\frac{d}{dx} b^x = b^x \ln b.
This holds everywhere, with the factor \ln b determining growth rate (positive if b > 1, negative if $0 < b < 1).[43]