Re: base e: making negative exponents positive
From: Michael Stemper (mstemper_at_siemens-emis.com)
Date: 02/08/05
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Date: Tue, 8 Feb 2005 16:07:23 -0600
In article <1107898827.565319.211470@l41g2000cwc.googlegroups.com>, davidcsnow writes:
>is the following possibe? and if so, why? I havent looked at this stuff
>in a while, so its a bit confusing...and help would be great!
>
>Start with:
>
>e (to the power of) -5/t = 0.20
>
>Then I have to change '-5/t' to '+5/t' giving:
>
>e (to the power of) +5/t = 5.00
>
>I just dont understand, and cant find the rules that explain why 0.20
>becomes 5.00 when you make the negative exponent positive.
It's the laws of exponents. Let's start with integer exponents. (Keep in
mind that "x squared" is usually written as "x^2".)
x^3 = x*x*x, right?
(x^3)/x = x^2 = x*x, right?
(x^2)/x = x^1 = x, right?
This is pretty simple so far (I hope). To keep things consistent,
mathematicians then defined (x^1)/x = x^0 = 1. Do you see that this
makes some sense? Each time that we divide by x, we reduce the exponent
by one.
The next law of exponents to think about is that x^(a+b) = (x^a)*(x^b).
For instance, x^3 = x^(1+2) = (x^1)*(x^2) = x*x*x. Do you buy that? If
so, then to keep things consistent, define:
x^(-y) = 1/(x^y)
Why is this consistent? Because if we take (x^y)*(x^(-y)), we get
x^(y+(-y)), or x^0, which is 1. So, changing the sign of the exponent
*means* taking the reciprocal.
(One other note: everything that I've said above only works for positive
values of x. The exponents can be positive, negative, zero, integer,
rational, or real. The base has to be positive, or things get much more
complicated. In your example, the base was "e", which is a positive
number, so we're OK.)
-- Michael F. Stemper #include <Standard_Disclaimer> Visualize whirled peas!
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