-1

How do I solve this equation for P? For everything I've tried, P ends up trapped in an exponent or another natural log.

$$ \ln\left(\frac{GC}{a}\right) = hP + \ln(P) $$

IV_
  • 6,964
Koryak21
  • 1
  • 1
  • 1
    There isn't an elementary solution. You need to use things like Lambert's W function. – Arturo Magidin Dec 01 '22 at 19:36
  • We cannot rearrange the equation for $P$ by only applying only finite numbers of only elementary functions/operations to the equation. The reason is described in https://math.stackexchange.com/questions/4586412/how-can-we-show-that-az-ez-and-alnz-z-have-no-elementary-inverse/4586413#4586413 – IV_ Dec 01 '22 at 20:17

1 Answers1

3

Solution will involve the Lambert W function.

The title question is the definition: $$ x + \ln x = \ln c \quad\Leftrightarrow\quad xe^x = c \quad\Leftrightarrow\quad x = W(c) $$ For the more complicated question... $$ \ln\left(\frac{GC}{a}\right) = hP + \ln(P) \\ \frac{GC}{a} = Pe^{hP} \\ \frac{hGC}{a} = hPe^{hP} \\ W\left(\frac{hGC}{a}\right) = hP \\ \frac{1}{h}W\left(\frac{hGC}{a}\right) = P $$

GEdgar
  • 111,679