Exponential Function Instance

An exponential function instance is a continuous algebraic function from an exponential function family (based on the exponentiation operation of the form [math]\displaystyle{ f(x,b)=b^x }[/math], where [math]\displaystyle{ b }[/math] is the exponent base and [math]\displaystyle{ x }[/math] is the exponent).

• Context:
  • It can (often) assume that the Exponent Base is Euler's Number ([math]\displaystyle{ e }[/math]).
  • It can range from being a Real Exponential Function to being a Complex Exponential Function.
  • It can be a member of an Exponential Equation.
  • It can be an input to an Exponential Operation, such as [math]\displaystyle{ f(x,a) \times f(x,b) = f(x,a+b) }[/math].
• Example(s):
  • [math]\displaystyle{ f(x,10)=10^x }[/math], with base of 10, and exponent x.
  • [math]\displaystyle{ f(x,2)=2^x }[/math], with base of 2, and exponent x.
  • [math]\displaystyle{ f(x,\empty)=e^x }[/math], with exponent base of e (a Gaussian function).
  • a Fourier Transform.
  • a Logistic Curve Function: [math]\displaystyle{ f(t,A,B,C) \equiv (C + Ae^{-Bt})^{-1} }[/math].
  • …
• Counter-Example(s):
  • a Linear Function.
  • a Polynomial Function.
  • a Log Function.
• See: Natural Exponential function, Exponential Distribution, Logarithm Function, Exponential Polynomial Function, Base (Exponentiation), E (Mathematical Constant), Transcendental Number, Derivative.

---

---

References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/exponential_function Retrieved:2015-11-9.
  • In mathematics, an exponential function is a function

    of the form : [math]\displaystyle{ f(x) = b^x. \, }[/math] The input variable x occurs as an exponent – hence the name. A function of the form ƒ(x) = b^x ± c is also considered an exponential function, and a function of the form ƒ(x) = a·b^x can be re-written as ƒ(x) = bx ± c by the use of logarithms and so is an exponential function.

    In contexts where the base b is not specified, especially in more theoretical contexts, the term 'exponential function is almost always understood to mean the natural exponential function : [math]\displaystyle{ x \mapsto e^x, \, }[/math] where e is Euler's number, a transcendental number approximately 2.718281828. The reason this number e is considered the "natural" base of exponential functions is that this function is its own derivative. Every exponential function is directly proportional to its own derivative, but only when the base is e does the constant of proportionality equal 1. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. The function is often written as exp(x), especially when it is impractical to write the independent variable as a superscript. The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics. The graph of is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis but can get arbitrarily close to it for negative x ; thus, the x-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y coordinate at that point. The inverse function is the natural logarithm ln(x); because of this, some old texts refer to the exponential function as the antilogarithm.

    In general, the variable x can be any real or complex number or even an entirely different kind of mathematical object; see the formal definition below.

---