How to change the subject when exponential is involved

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The discussion focuses on rearranging the equation T = S [(1-Pr(x))^N] + Pr(x) to make Pr(x) the subject. It is established that for N ≥ 5, the equation becomes a polynomial of at least the 5th degree, which typically lacks analytic solutions. Participants suggest that numerical solutions and approximations are necessary, and mention the Lambert W function as a potential method for expressing the solution, despite its impracticality for standard calculators.

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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!
 
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Sorry, I should mention N isn't necessarily an integer.
 
MWD02 said:
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 said:
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)
 
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
 
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!
 

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