How to solve $(a+\sqrt{b})^n - (a-\sqrt{b})^n = x$? Consider equation $(a+\sqrt{b})^n - (a-\sqrt{b})^n = x$
How do I properly solve for $n$ given $x$?
 A: There are many approaches.  If you think $n$ is a small whole number, you can just try a range.  A spreadsheet would make this easy.  If one of $a+\sqrt b$ or $a-\sqrt b$ is smaller than $1$ it will to to zero as $n$ increases.  To be definite, assume $a-\sqrt b$ is smaller than $1$.  Let's ignore it for a moment.  Then $n \approx \frac {\log x}{\log (a+\sqrt b)}$ where you can use your favorite base for the logs.  You can use numerical methods.  The left side will be monotonic with $n$, so any reasonable root-finder will work.  Just graphing will get you very close.
As an example, suppose you want to find the index of a Fibonacci number.  We are given that $\sqrt 5 F_n=(\frac {1+\sqrt 5}2)^n-(\frac {1-\sqrt 5}2)^n$.  As $|\frac {1-\sqrt 5}2|\lt 1$, powers of it go to zero quickly.  If somebody gives us $14930352$ and asks which number it is, we can just do $\frac {\log (14930352\sqrt 5)}{\log (\frac {1+\sqrt 5}2)}$ and get a number that is within $10^{-14}$ of $36$, so $n=36$
A: 
Consider equation $(a+\sqrt{b})^n - (a-\sqrt{b})^n = x$

Consult this link: The Binomial Theorem. 
Since you are given $x$ (taken to be a constant), you can also try take logarithms of each side of the equation.
