Simplifying equations while holding some constraints

Question

I have two equation and i tried to solve them but i can't simplify them :

1. $(\frac{1}{b})^{\frac{1}{r}} = 0.8$
2. $ br = n$

$r = \frac{n}{b} \rightarrow \frac{r}{n} = \frac{1}{b}$.

Plugging this equation in first one i have:

1. $(\frac{r}{n})^{\frac{1}{r}} = 0.8$

Now i have :

$\frac{r}{n} = {(0.8)}^{r}$

Now suppose i know the value of $n = 20:$

How can i derive the equation in terms of $r$?

Answer 1

You will need the Lambert W function for this, which is non-elementary ($W(xe^x)=x$): $$r/n=0.8^r=e^{r\ln0.8}$$ $$re^{-r\ln0.8}=n$$ $$re^{r\ln1.25}=n$$ $$(r\ln1.25)e^{r\ln1.25}=n\ln1.25$$ $$r\ln1.25=W(n\ln1.25)$$ $$r=\frac{W(n\ln1.25)}{\ln1.25}$$ If $n=20$ then $r=5.658289\dots$