Trouble simplifying a tough equation

Question

I'm having trouble simplifying the following equation. I've tried grouping terms in different ways, but it's not looking any more joyful. Can someone please help with its resolution? Hopefully by midnight?? $$ \omega - \ln Y=\ln\left(H p^2 a + \exp(ra)\right) - \ln N $$

Answer 1

$$ \omega - \ln Y=\ln\left(H p^2 a + \exp(ra)\right) - \ln N $$ $$ \omega =\ln\left(H p^2 a + \exp(ra)\right)+ \ln Y - \ln N $$ $$ \omega =\ln\left(H p^2 a + \exp(ra)\right)+ \ln (Y/N ) $$ $$ \omega =\ln\left({YH p^2 a + Y\exp(ra)\over N}\right) $$ $$ \omega =\ln\left({Ha p^2Y + Ye^{ar})\over N}\right) $$ $$e^{\omega}=\frac{Ha p^2Y + Ye^{ar}}{N}$$ $$0=Ha ppY-Ne^{\omega} + Ye^{ar}$$

Answer 2

Assuming everything there are numbers (or at least, everything commutes), may be you are wishing the community "$HappY Ne^\omega Ye^{ar}$".

Answer 3

Here, I solved for $a$ for you:

$$ a = \frac{N r e^\omega-H p^2 Y \cdot \mathrm{W}\left(\frac{e^{\frac{e^\omega N r}{H p^2 Y}} r}{H p^2}\right)}{H p^2 r Y} $$

Where $\mathrm{W}(x)$ is the Lambert-W function.