Exponential Equations

The natural exponential function along part of the real axis

Exponential functions with bases 2 and 1/2

The exponential function is a mathematical function denoted by $f(x)=\exp(x)$ or $e^{x}$ (where the argument x is written as an exponent). It can be defined in several equivalent ways. Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". Its value at 1, $e=\exp(1),$ is a mathematical constant called Euler's number.

The exponential function equals its own derivative. Thus, it appears in the solutions of many differential equations.

Moreover, it satisfies the identity

$e^{x+y}=e^{x}e^{y}{\text{ for all }}x,y\in \mathbb {C} ,$

which, along with the definition $e=e^{1}$, shows that $e^{n}=\underbrace {e\times \cdots \times e} _{n{\text{ factors}}}$ for positive integers n, relating the exponential function to the elementary notion of exponentiation.

The argument of the exponential function can be any real or complex number. This allows one to extend the concept of exponentiation (which would normally be defined only for integer exponents as: $a^{n}=\underbrace {a\times \cdots \times a} _{n{\text{ factors}}}$ for integer n) to real or complex exponents, by defining $a^{x}:=\exp(x\ln(a))$ for positive a and real or complex x. The exponential of a complex argument is closely related to trigonometry as shown by Euler's formula. The argument can even be an entirely different kind of mathematical object (for example, a square matrix).

In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences, as in a self-reproducing population, a fund accruing compound interest, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within physics, computer science, chemistry, engineering, mathematical biology, and economics.

The real exponential function is a bijection from $\mathbb {R}$ to $(0;\infty )$. Its inverse function is the natural logarithm, denoted $\ln ,$ $\log ,$ or $\log _{e};$ because of this, some old texts refer to the exponential function as the antilogarithm.

The exponential function is sometimes called the natural exponential function for distinguishing it from the other exponential functions, which are the functions of the form $f(x)=ab^{x},$ where the base b is a positive real number. The definition $b^{x}\ {\stackrel {\text{def}}{=}}\ e^{x\ln b}$ for positive b and real or complex x establishes a strong relationship between these functions, which explains this ambiguous terminology.

Graph

The graph of $y=e^{x}$ is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation ${\tfrac {d}{dx}}e^{x}=e^{x}$ means that the slope of the tangent to the graph at each point is equal to its y-coordinate at that point.

Relation to more general exponential functions

The exponential function $f(x)=e^{x}$ is sometimes called the natural exponential function for distinguishing it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since per definition, for positive b,

$ab^{x}:=ae^{x\ln b}$

As functions of a real variable, exponential functions are uniquely characterized by the fact that the derivative of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b:

${\frac {d}{dx}}b^{x}\ {\stackrel {\text{def}}{=}}\ {\frac {d}{dx}}e^{x\ln(b)}=e^{x\ln(b)}\ln(b)=b^{x}\ln(b).$

For b > 1, the function $b^{x}$ is increasing (as depicted for b = e and b = 2), because $\ln b>0$ makes the derivative always positive; while for b < 1, the function is decreasing (as depicted for b = ${\tfrac {1}{2}}$); and for b = 1 the function is constant.

Euler's number e = 2.71828... is the unique base for which the constant of proportionality is 1, since $\ln(e)=1$, so that the function is its own derivative:

${\frac {d}{dx}}e^{x}=e^{x}\ln(e)=e^{x}.$

This function, also denoted as exp x, is called the "natural exponential function", or simply "the exponential function". Since any exponential function can be written in terms of the natural exponential as $b^{x}=e^{x\ln b}$, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by

$x\mapsto e^{x}$ or $x\mapsto \exp x.$

The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is a complicated expression.

For real numbers c and d, a function of the form $f(x)=ab^{cx+d}$ is also an exponential function, since it can be rewritten as

$ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}.$

Formal definition

The exponential function (in blue), and the sum of the first n + 1 terms of its power series (in red).

The real exponential function $\exp \colon \mathbb {R} \to \mathbb {R}$ can be characterized in a variety of equivalent ways. It is commonly defined by the following power series:

$\exp x:=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+\cdots$

Since the radius of convergence of this power series is infinite, this definition is, in fact, applicable to all complex numbers $z\in \mathbb {C}$. The constant e can then be defined as ${\textstyle e=\exp 1=\sum _{k=0}^{\infty }(1/k!).}$

The term-by-term differentiation of this power series reveals that ${\frac {d}{dx}}\exp x=\exp x$ for all real x, leading to another common characterization of $\exp x$ as the unique solution of the differential equation

$y'(x)=y(x),$

satisfying the initial condition $y(0)=1.$

Based on this characterization, the chain rule shows that its inverse function, the natural logarithm, satisfies ${\frac {d}{dy}}\log _{e}y=1/y$ for $y>0,$ or ${\textstyle \log _{e}y=\int _{1}^{y}{\frac {1}{t}}\,dt.}$ This relationship leads to a less common definition of the real exponential function $\exp x$ as the solution $y$ to the equation

$x=\int _{1}^{y}{\frac {1}{t}}\,dt.$

By way of the binomial theorem and the power series definition, the exponential function can also be defined as the following limit:

$\exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.$

It can be shown that every continuous, nonzero solution of the functional equation $f(x+y)=f(x)f(y)$ is an exponential function, $f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto e^{kx},$ with $k\in \mathbb {R} .$

Overview

The red curve is the exponential function. The black horizontal lines show where it crosses the green vertical lines.

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 to the number

$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$

now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.

If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the interest earned each month is ${\tfrac {x}{12}}$ times the current value, so each month the total value is multiplied by ($1+{\tfrac {x}{12}}$), and the value at the end of the year is ($1+{\tfrac {x}{12}}$)¹². If instead interest is compounded daily, this becomes ($1+{\tfrac {x}{365}}$)³⁶⁵. Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function,

$\exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}$

first given by Leonhard Euler. 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,

$\exp(x+y)=\exp x\cdot \exp y$

which justifies the notation e^x for exp x.

The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself (rather than equal to it) is expressible in terms of the exponential function. This function property leads to exponential growth or exponential decay.

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 also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra.

Resources

• Exponential and Logarithmic Equations, Book Chapter
• Guided Notes

Licensing

Content obtained and/or adapted from:

• Exponential function, Wikipedia under a CC BY-SA license