How to change the subject when exponential is involved

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Discussion Overview

The discussion revolves around the challenge of making Pr(x) the subject of the equation T = S [(1-Pr(x))^N] + Pr(x). Participants explore whether this is possible, especially considering that N is not necessarily an integer.

Discussion Character

  • Homework-related, Mathematical reasoning, Exploratory

Main Points Raised

  • One participant expresses uncertainty about whether it is possible to isolate Pr(x) in the given equation and requests assistance.
  • Another participant notes that if N is an integer and particularly for N ≥ 5, the equation becomes a polynomial of at least the 5th degree, which typically lacks analytic solutions.
  • A different participant mentions that numerical solutions may be necessary, implying the need for approximations.
  • One participant suggests that the equation could potentially be rearranged to involve the Lambert W function, although they acknowledge that most calculators do not support this function.

Areas of Agreement / Disagreement

Participants generally agree that isolating Pr(x) analytically may not be feasible, particularly for higher values of N, but there is no consensus on the best approach to take or the implications of using numerical methods versus the Lambert W function.

Contextual Notes

Participants highlight the limitations of the problem, including the dependence on the nature of N and the potential need for approximations in finding solutions.

MWD02
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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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