MHB How to change the subject when exponential is involved

[MWD02]

It's been a long time since I've worried about this - but could someone help me make Pr(x) the subject (I can't remember if it's possible, if it's not, I'd love a brief explanation):

T = S [(1-Pr(x))^N] + Pr(x)

Thanks in advance!

 

[MWD02]

Sorry, I should mention N isn't necessarily an integer.

 

[I like Serena]

It's been a long time since I've worried about this - but could someone help me make Pr(x) the subject (I can't remember if it's possible, if it's not, I'd love a brief explanation):

T = S [(1-Pr(x))^N] + Pr(x)

Thanks in advance!

Sorry, I should mention N isn't necessarily an integer.

Hi MWD02! Welcome to MHB! ;)

Even if $N$ would be an integer, for $N\ge 5$ this is a polynomial of at least the 5th degree with at least 3 terms, for which there are typically no 'analytic' solutions.
So I think we're stuck with numerical solutions, meaning we have to make approximations. (Worried)

 

[MWD02]

Ah I was afraid someone would use the word "approximations"!... Oh well, I'll see what I can do for my particular problem.

Thanks very much for the reply! :D

 

[HallsofIvy]

Well, you probably could (I won't try to do it) rearrange the equation so the solution can be written in terms of the "Lambert W function" (defined as the inverse function to f(x)= xe^x) but then your calculator probably does not have a "W function" key!