# Solving an equation on a particular domain [closed]

I am trying to solve this equation for $\alpha$: $n=1+\frac{1}{ln(\alpha)}[ln\frac{w}{P-r}+ln\frac{s}{s-\beta}+ln(\frac{lns}{ln(\alpha s)})]$ with the following constraints: $\beta<s$ and $\alpha s>1$.
My code looks like this:

Solve[{-1 + n - (-Log[P - r] + Log[s] - Log[-b + s] + Log[w] + Log[Log[s]] - Log[Log[a s]])/Log[a] == 0, a*s > 1, b < s}, a]


I've also tried

Solve[-1 + n - (-Log[P - r] + Log[s] - Log[-b + s] + Log[w] + Log[Log[s]] - Log[Log[a s]])/Log[a] == 0, a]


which is also not working.

I keep on getting "This system cannot be solved with the methods available to Solve."

Can someone point me to the right direction?

Thanks!

UPDATE: Thanks for pointing out my syntax errors. I have replaced ln with Log. Still not working. It seems like there is no closed-form solution, wondering if there's any way to get an approximation, especially since I have actual values for all the variables ${n,s,P,r,w,\beta}$.

• Use Log, not ln – Carl Woll Apr 20 '17 at 20:51
• Hi; I followed Carl's advice of course and corrected the obvious syntax errors and it is running for 10 minutes. If you are expecting some answer like a -> Log[2]/w it just does not seem to be in the cards. If Mma gets the answer it may be 20 pages long and therefore of no value to us humans. Having a CAS does not mean shutting your brain down. You have to prepare the problem beforehand for solution. You have to bound those constants or give them values, in other words all that needs to be cleaned up and simplified. – bobbym Apr 20 '17 at 21:34
• Carl said Log, not log. Look it up in the documentation. – Michael E2 Apr 20 '17 at 23:12
• Sorry, I probably should have also added that I am looking to run this on a dataset where I have the value for every variable EXCEPT $\alpha$ which is the reason why I am looking for a closed-form solution. All variables are positive. @CarlWoll I think that goes towards what you were saying. – Lnmx Apr 20 '17 at 23:27
• Log is not the same as log. – Daniel Lichtblau Apr 20 '17 at 23:33

## 1 Answer

It looks like Solve can produce a solution for explicit integer values of n. For example:

Solve[3-(-Log[P-r]+Log[s]-Log[-b+s]+Log[w]+Log[Log[s]]-Log[Log[a s]])/Log[a]==0,a]//TeXForm


Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

$\left\{\left\{a\to \frac{\sqrt[3]{-3} \sqrt[3]{s} \sqrt[3]{w} \sqrt[3]{\log (s)}}{\sqrt[3]{b-s} \sqrt[3]{P-r} \sqrt[3]{W\left(\frac{s w \log (s) \left(3^{\frac{r s}{3 b P}-\frac{s}{3 b}-\frac{r}{3 P}+\frac{1}{3}} s^{\frac{r s}{b P}-\frac{s}{b}-\frac{r}{P}+1}\right)^{\frac{3 b P}{(b-s) (P-r)}}}{(s-b) (P-r)}\right)}}\right\},\left\{a\to -\frac{\sqrt[3]{3} \sqrt[3]{s} \sqrt[3]{w} \sqrt[3]{\log (s)}}{\sqrt[3]{b-s} \sqrt[3]{P-r} \sqrt[3]{W\left(\frac{s w \log (s) \left(3^{\frac{r s}{3 b P}-\frac{s}{3 b}-\frac{r}{3 P}+\frac{1}{3}} s^{\frac{r s}{b P}-\frac{s}{b}-\frac{r}{P}+1}\right)^{\frac{3 b P}{(b-s) (P-r)}}}{(s-b) (P-r)}\right)}}\right\},\left\{a\to -\frac{(-1)^{2/3} \sqrt[3]{3} \sqrt[3]{s} \sqrt[3]{w} \sqrt[3]{\log (s)}}{\sqrt[3]{b-s} \sqrt[3]{P-r} \sqrt[3]{W\left(\frac{s w \log (s) \left(3^{\frac{r s}{3 b P}-\frac{s}{3 b}-\frac{r}{3 P}+\frac{1}{3}} s^{\frac{r s}{b P}-\frac{s}{b}-\frac{r}{P}+1}\right)^{\frac{3 b P}{(b-s) (P-r)}}}{(s-b) (P-r)}\right)}}\right\}\right\}$

If you look at the answer for various values of n, you may be able to guess what the general solution might be.

PS Solve is completely unable to solve transcendental equations involving an unknown function ln