constant e

If I recall correctly Bernoulli discovered the number e when he was studying compound interest.

Using the compound interest formula let the the number of times the amount get computed go to infinity
 
Rahul singh india said:
How the value of constant e was derived by jacob bernoulli ?
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This reply is late and not about Bernoulli. Leonard Gillman was a giant of the 20th century's calculus reform. In addition to his (with Jerison) ground breaking work in rings of continuous functions, he was president of the American Mathematics Association. So to relate to to the O.P. : In the Gillman's first edition of Calculus (1973), they begin intruding the logarithm function is a different way.
The most widely recognized property is log⁡(x⋅y)=log⁡(x)+log⁡(y)\displaystyle \log(x\cdot y)=\log(x)+\log(y).
Thus in the text they define a function x>0,L(x)=∫1x1tdt\displaystyle x > 0,\quad L(x) = \int_1^x {\frac{1}{t}dt} .
So look at α,β>0,L(αβ)=∫1αβ1tdt=∫1α1tdt+∫ααβ1tdt\displaystyle \alpha ,\beta > 0,\quad L(\alpha \beta ) = \int_1^{\alpha \beta } {\frac{1}{t}dt} = \int_1^\alpha {\frac{1}{t}dt} + \int_\alpha ^{\alpha \beta } {\frac{1}{t}dt}
Using u-substitution\displaystyle u\text{-substitution} with ∫ααβ1tdt\displaystyle \int_\alpha ^{\alpha \beta } {\frac{1}{t}dt} If βu=t\displaystyle \beta u=t we get ∫1α1tdt+∫1β1udu=L(α)+L(β)\displaystyle \int_1^\alpha {\frac{1}{t}dt} + \int_1^\beta {\frac{1}{u}du}=L(\alpha)+L(\beta)
There is that basic property. Moreover, the derivative is L′(x)=1x\displaystyle L'(x)=\frac{1}{x} just as it should be.
Thus that text defines log⁡(x)=L(x)=∫1x1tdt\displaystyle \large\bf{\log(x)=L(x)=\int_1^x {\frac{1}{t}dt}}
The number e\displaystyle \bf{e} is the number for which L(e)=1\displaystyle \bf L(e)=1
 
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