How do I solve this equation for P? For everything I've tried, P ends up trapped in an exponent or another natural log.
$$ \ln\left(\frac{GC}{a}\right) = hP + \ln(P) $$
2026-04-02 15:42:33.1775144553
How to solve for $x$ from $x + \ln(x) = \ln(c)$?
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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... $$ \ln\left(\frac{GC}{a}\right) = hP + \ln(P) \\ \frac{GC}{a} = Pe^{hP} \\ \frac{hGC}{a} = hPe^{hP} \\ W\left(\frac{hGC}{a}\right) = hP \\ \frac{1}{h}W\left(\frac{hGC}{a}\right) = P $$