What Makes the Number e So Special in Logarithmic Functions?

[JPC]

hey

i know what the log function is

log a (x) : can be translated to :
a ^ y = x
with y to find

but with the Ln its
ln(x) = e ^ y = x
with y to find

and e = about 2.7...

But why this number ?
whats special about 2.7... ?
whats the exact value of e ?

 

[arildno]

The exact value of e is e.

Strategies to convert that value into its equivalent decimal representation abound, the most common centering on the defining identity of e:
$$e=\sum_{n=0}^{\infty}\frac{1}{n!}, 0!=1, n!=n*((n-1)!), n\geq{1}$$

 

[Ahmed Ismail]

Goodmoring ALL,
the number e as "arildno" e=e and e can be calculated by an infinite series and it is a Real number.
But you asked why e?
we must know that e is the contraction of eulur, and was a mathematics scientist and he had make a great researches aboiut the number e.
also, what special about e?
physicains had found many phenomenans the changes by a function f(x) = log a(x) [log to the base a]
and the found that the approximate value of a =2.7...

 

[tehno]

and e = about 2.7...

But why this number ?
whats special about 2.7... ?
whats the exact value of e ?

Ever heard that $ln$ is sometimes called a natural logarithm.
Why would they call that natural and what a heck does $e$ have to do with nature you may ask.
Instead of answering this in a long and bit a philosophical way ,I will ask you quite a similar question :Why $\pi$ ,and why is important about that constant.I will not answer neither of two questions leaving you to ponder over them alone.
If you ask why is $e=2.71...$and $\pi=3.14...$ than the answer is simple:That's becouse people like to use decimal number system (and I guess I know why ,when I take a look at my hands).
The last question "what's the exact value of e"?.The most difficult one.
Well I think I will not dare answering that one...

 

[arildno]

I've dared:
The exact value of e is e.

 

[mathwonk]

the exact value of e is: the smallest real number larger than all finite sums of the series:

1 + 1 + 1/2! + 1/3! + 1/4! + 1/5!+....

it is also exactly the unique x value such that the area under the graPH OF Y = 1/X, FROM X=1 TO x=e IS 1.

it is also probably the unique smallest real number larger than all the powers of form (1+ 1/n)^n, for all positive integers n.

it is approximately equal to 2.718281828459...

e is the unique positive real number a such that the derivative of the function a^x is a^x.

e is the value at x = 1, of the unique solution of the equation

f' = f, f(0) = 1.

e is the unique positive real number a such that the function f(x) = a^x has derivative at zero equal to 1, i.e. such that f'(0) = 1.

 

[tehno]

I've dared:
The exact value of e is e.

Yes ,yes .Of course,that is an alphabetical number system !
How could I forget that one...

 

[arildno]

Two point seven, two times Ibsen, 459045...

 

[D H]

Why e?

"Real and Complex Analysis" by Walter Rudin starts with a prologue on the exponential function. The first sentence is "The exponential function is the most important function in mathematics."

Among other very useful features,

• The exponential function is the only function (to within a multiplicative constant) whose derivative is equal to itself.
• The exponential function is related to the trigonometric functions via the Euler formula,$\exp(ix) = \cos x + i\sin x$.
• The derivative of the inverse of the exponential function, $\log x$, is simply $1/x$.

 

[Eighty]

The natural logarithm ln(x) is the simplest of the logarithms. It's defined as the area (or integral, if you will) under the graph y=1/t when t goes from 1 to x. Specifically, when the area is 1, ln(x)=e (just like mathwonk said). Also, the inverse function of ln(x) is e^x, which you may know is the only function which is its own derivative (well, it can be multiplied by a constant too), which makes it even more special.

 

[JeffKoch]

http://en.wikipedia.org/wiki/Eulers_number

 

[arildno]

Euler himself used "e" as signifying the exponential function.
Only fifth-rate mathematicians assign their own name to their objects of study, and Euler was top-of-the-notch.

Noone knows why he called the number "e"; it is perhaps most likely it originally was short-hand for "that number which serves most "naturally" as the base in Exponential functions".

 

[HallsofIvy]

Any exponential function a^x has the property that its derivative (rate of change) is just a constant times itself: Ca^x. You can show that for a= 2, that number is less than 1 but that for a= 3 it is larger than 1. There exist a number between 2 and 3 such that the constant is exactly 1. That is the number we call e. The function e^x has the property that its derivative is simply e^x itself.

 

[JPC]

the exact value of e is: the smallest real number larger than all finite sums of the series:

1 + 1 + 1/2! + 1/3! + 1/4! + 1/5!+....

What does the : '!' mean ?

 

[arildno]

See post 2, for definition of n!

 

[cristo]

"!" means factorial: 5!=5*4*3*2*1. So, in general n!=n(n-1)(n-2)...1.

 

[JPC]

thanks

i tested with a program :
with ent.text : sum of all from 0 to ent
res.tex : the result

Dim c As String
Dim d As String
Dim f As String
Dim a As String

f = "1"
a = "1"

c = 1

Do Until c * 1 = ent.Text * 1
a = a * 1 * c * 1
d = 1 / a * 1 f = f + d * 1

c = c + 1 * 1

Loop

res.Text = f
and result is good