I would like to solve this equation:
$$n \cdot 2^n = 15000$$
And according to WolframAlpha
$$n=\frac{W(15000\log(2))}{\log(2)}, \text{ where }\log\text{ is ln}$$
Which shows that I need to use the product log function $W$ which I tried looking up on wikipedia. I don't need the complex numbers, just real numbers.
Additionally, are there ways of solving the original equation without the $W$ functions?
Can someone explain the rules and how to do this please?
I would eventually like to implement a way to find $n$ programmaticly (if possible-in python).
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$\begingroup$You can use Newton's method for finding the root of $f(n) = n 2^n - 15000 = 0$: $$ n_{i+1} = n_i - \frac{f(n_i)}{f'(n_i)} = n_i - \frac{n_i 2^{n_i} - 15000}{2^{n_i} (1+n_i \log(2))} $$ and start with a suitable guess (look up the Wikipedia page on that). Here $n_0 = 1$ should do.You keep repeating the process for $i \ge 1$ until you reach a proper accuracy, i.e., you terminate at iteration $t$ when $n_{t} - n_{t-1} < \epsilon$ for some tiny $\epsilon$ you define.
Questions on exact implementation are more suitable question for SO; also scicomp.SE in case you're interested in details on appropriate numerical methods, and robust libraries.
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