As I've said before in other questions I've posed here, I'm an armchair mathematician and don't fully grasp a lot of more advanced concepts (I cap out around early AB Calc), and a lot of my knowledge/interest in maths comes from Numberphile, Mathologer, and other channels like that.
Anyway, I found a formula for the Fibonacci sequence: $$F(x)=\frac{\phi^x-\phi^{-x}\cos(x\pi)}{\sqrt{5}}$$
Out of curiosity, I wanted to find its inverse so that I can find values of the Fibonacci sequence that equate certain numbers (viz, $\pi$, $\tau$, $\phi$, &c).
To do this, I changed $F(x)$ to $y$ and then just flipped the variables, giving: $$x=\frac{\phi^y-\phi^{-y}\cos(y\pi)}{\sqrt{5}}$$
Obviously, the first step is to multiply both sides of the equation by $\sqrt{5}$ yielding: $$x\sqrt{5}=\phi^y-\phi^{-y}\cos(y\pi)$$
It's here that I'm stuck. My instinct tells me to take the $\ln$ of both sides to get the $y$'s to drop down yielding (I think): $$\ln\left(x\sqrt{5}\right)=y\ln(\phi)+y\ln(\phi)\cos(y\pi)$$
but I'm also concerned about the $\cos(y\pi)$ because (a) I don't know if that would be included in the $\ln$ function and (b) because I don't know how to get rid of it and if that would complicate getting the $y$'s out of the exponents.
Is there even a way to solve for $y$, or is it condemned to be in terms of $x$?