- #1

Loren Booda

- 3,119

- 4

e

[inte] dx/x=1, etc."

1

In other words,

__why__choose this function to define e, and how does it most fundamentally relate to other uses of e?

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- Thread starter Loren Booda
- Start date

- #1

Loren Booda

- 3,119

- 4

e

[inte] dx/x=1, etc."

1

In other words,

- #2

Brad_Ad23

- 502

- 1

That's a good question. I'm sure I could answer it where it not so late, I'll think into it.

- #3

e = lim (1 + x) ^(1/x)

where limit takes x to zero.

Instanton

- #4

Brad_Ad23

- 502

- 1

One thing that I remember is that for the graph of e

- #5

Instanton

- #6

newton1

- 152

- 0

it's natural log

integrate from 1 to x for dt/t = ln(x)-ln(1)= ln(x)

because after we define natural logarithm funtion

we fine the base for the ln function

so we defined the exponatial e

hope this can help

- #7

rate of change of X proportional to X

eg X'(t) = c*X(t). For example this very often occurs when X is a number of objects which are independent of one another: reproducing bacteria, decaying atoms. The most mathematically natural case is where c=1; though physically, there is nothing special about that unless the units of X and t are comparable.

It turns out the solution to this differential equation has the form of an exponential, X(t)=e^t, with e itself.

- #8

- #9

In this form it occurs also in other scenarios like the calculation of continuous interest in finance...

As far as considering it as a natural base for logarithms... you realize how natural it is only once you start calculating derivatives I guess...

Another possibility is to say that e is the basis of logarithm when

lim (log(1+x))/x=1

x->0

that is the logarithm that goes to zero as linearly as x does.

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