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araku valley Grade: 11
        

1.WHY "e" IS SO SPECIAL?WHY ITS NATURAL?


2.WHY LOG BASE e CALLED NATURAL LOGARITHM?


WHATS SO NATURAL ABOUT IT?PLEASE REPLY.

7 years ago

Answers : (1)

Askiitians Expert Himanshu Geed - IIT Delhi
31 Points
										

Dear Araku,


1. e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is exactly 1. he number e is irrational; it is not a ratio of integers. Furthermore, it is transcendental; it is not a root of any non-zero polynomial with rational coefficients.




2. loge is a "natural" log because it automatically springs from, and appears so often in, mathematics. For example, consider the problem of differentiating a logarithmic function:


\frac{d}{dx}\log_b(x) = \frac{\log_b(e)}{x}  =\frac{1}{x\ln(b)}

If the base b equals e, then the derivative is simply 1/x, and at x = 1 this derivative equals 1. Another sense in which the base e logarithm is the most natural is that it can be defined quite easily in terms of a simple integral or Taylor series and this is not true of other logarithms.


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Regards,

Askiitians Experts
Himanshu Geed

7 years ago
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