Re: why we take ln(e)=1?

In sci.math, Ron Sperber
<ronsperber@xxxxxxxxxxxxxxxx>
wrote
on Sat, 07 Apr 2007 19:04:22 -0400
<4618237d$0$5794$4c368faf@xxxxxxxxxxxxxx>:
The Ghost In The Machine wrote:
In sci.math, sufyan.iiui@xxxxxxxxx
<sufyan.iiui@xxxxxxxxx>
wrote
on 6 Apr 2007 05:10:56 -0700
<1175861456.658419.55620@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>:
i m mathematics student bt i really dont know why this function i mean
(exponential function)is so common.
u just reply ur openions.
bye


If memory serves, log(x) was originally defined as

log(x) = integral(u=1,x) (1/u du)

and

exp(x) = sum(i=0,+oo) (x^i / i!)

Originally defined by who?

I for one would guess Euler, but would have to look. 'e' turns out
to be his number. :-) In any event, my memory's not the best,
and it turns out ln(x) = log_e(x) and exp(x) = e^x.

When I teach Calc II I define ln(x) by the integral you mention, but
then define exp(x) as the inverse of ln(x). Then one can define e as the
unique number for which ln(e)=1. Then you can prove easily enough that
for a rational number x, exp(x)=e^x. This leads to a definition of e^x
as exp(x). Then one can show that e^x is equal to the power series you give.

It turns out these are inverses, since

d(log(x)) / dx = 1/x

pretty much by definition, and if one works out
each term (after proving exp(x) is absolutely convergent,
which isn't that difficult), one can easily see
that

d(exp(x)) / dx = exp(x)

Put more clearly, if

y = log(x)

then dy/dx = 1/x, and if

x = exp(y)

then dx/dy = exp(y) = x.

Granted, this is an abuse of the Leibnitz notation, but it
shows that they are an inverse up to a constant of integration.

That constant turns out to be zero, as one can
easily see since

exp(0) = 1

and

log(1) = 0.

I could be wrong, admittedly, but that's what I remember. :-)



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Relevant Pages

  • Re: why we take ln(e)=1?
    ... is so common. ... u just reply ur openions. ... shows that they are an inverse up to a constant of integration. ... That constant turns out to be zero, ...
    (sci.math)
  • Re: why we take ln(e)=1?
    ... u just reply ur openions. ... If memory serves, ... It turns out these are inverses, ... That constant turns out to be zero, ...
    (sci.math)