Jump to content

Wikipedia:Reference desk/Archives/Mathematics/2010 April 5

From Wikipedia, the free encyclopedia
Mathematics desk
< April 4 << Mar | April | May >> April 6 >
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.


April 5

[edit]

Determining x in exponentials

[edit]

Hello! I'm trying to find a way of determining the value of x in an equation of the type:

a*e^(b x)+b*e^(c x)+c*e^(d x)= k

Being a, b, c, d & k constants. Is there anyway for me to compute this? I was looking into logarithms, but maybe there is another way of doing this, considering that I cannot simplify a sum of logarithms...

Thanks for all the help 16:05, 5 April 2010 (UTC)~ —Preceding unsigned comment added by 193.137.208.122 (talk)

The solutions to the equation aebx+becx+cedx = k cannot be expressed in closed form, but you may expand the exponentials in taylor series to the second degree and get as an approximation the quadratic equation a(1+bx+b2x2/2)+b(1+cx+c2x2/2)+c(1+dx+d2x2/2) = k , which you can solve explicitly. See also root-finding algorithm. (It is unbelievable that b and c each occurs twice in the equation! ) Bo Jacoby (talk) 16:56, 5 April 2010 (UTC).[reply]

Letting u = ex, the equation

becomes

For some values of a, b, c, this can be solved in closed form. For example, if

b = 2,
c = 1,
d = 0,

then the whole thing is a quadratic equation. And if you have 3, 2, 1 instead of 2, 1, 0, then its cubic, but reduces to quadratic if k happens to be 0. Michael Hardy (talk) 20:18, 5 April 2010 (UTC)[reply]