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What is Exponential function mean?
In mathematics, the exponential function is the function f ( x ) = e x , {\displaystyle f(x)=e^{x},} where the base e = 2.71828... is Euler's number and the argument x occurs as an exponent. 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 ) = a b x , {\displaystyle f(x)=ab^{x},} where the base b is a positive real number. The formula b x = e x ln b {\displaystyle b^{x}=e^{x\ln b}} establishes a strong relationship between these functions, which explains this ambiguous terminology.
The exponential function equals its derivative and satisfies the identity
e x + y = e x e y for all x , y ∈ R . {\displaystyle e^{x+y}=e^{x}e^{y}{\text{ for all }}x,y\in \mathbb {R} .}There are several equivalent definition of the exponential function, some of them being based on the above properties.The argument of the exponential function can be any real or complex number, or even an entirely different kind of mathematical object (for example, a square matrix).
The ubiquitous occurrence of the exponential function in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". 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.
referencePosted on 22 Nov 2024, this text provides information on Miscellaneous in Academic & Science related to Academic & Science. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.
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