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In mathematics, an exponential function is a function of the form

The natural exponential function y = ex

Exponential functions with bases 2 and 1/2

{\displaystyle f(x)=ab^{x},}{\displaystyle f(x)=ab^{x},}

where b is a positive real number not equal to 1, and the argument x occurs as an exponent. For real numbers c and d, a function of the form {\displaystyle f(x)=ab^{cx+d}}{\displaystyle f(x)=ab^{cx+d}} is also an exponential function, since it can be rewritten as

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

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

{\displaystyle {\frac {d}{dx}}b^{x}=b^{x}\log _{e}b.}{\displaystyle {\frac {d}{dx}}b^{x}=b^{x}\log _{e}b.}

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

1

/

2

); and for b = 1 the function is constant.

The constant 1=e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function is its own derivative:

{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}\log _{e}e=e^{x}.}{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}\log _{e}e=e^{x}.}

This function, also denoted as exp x, is called the "natural exponential function",[1][2][3] or simply "the exponential function". Since any exponential function can be written in terms of the natural exponential as {\displaystyle b^{x}=e^{x\log _{e}b}}{\displaystyle b^{x}=e^{x\log _{e}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

{\displaystyle x\mapsto e^{x}}{\displaystyle x\mapsto e^{x}} or {\displaystyle x\mapsto \exp x.}{\displaystyle 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. The graph of {\displaystyle y=e^{x}}y=e^{x} is upward-sloping, and increases faster as x increases.[4] 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 {\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}}{\displaystyle {\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. Its inverse function is the natural logarithm, denoted {\displaystyle \log ,}{\displaystyle \log ,}[nb 1] {\displaystyle \ln ,}{\displaystyle \ln ,}[nb 2] or {\displaystyle \log _{e};}{\displaystyle \log _{e};} because of this, some old texts[5] refer to the exponential function as the antilogarithm.

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Step-by-step explanation: