I'm trying to make a function for the time a projectile is traveling over a certain distance, but while trying to solve t I noticed that t is both a power and a base, so use of logarithms would be basically useless. In the case of: $$t=\frac{x_{dist}*\cosθ}{\frac{v_0*\cosθ+v_0*\cosθ*(1-\frac{a_{drag\%}}{100})^t}{2}}$$ I went to the form: $$t=\frac{2*x_{dist}*\cosθ}{v_0*\cosθ+v_0*\cosθ*(1-\frac{a_{drag\%}}{100})^t}$$ Which can then be simplified to being: $$t=\frac{2*x_{dist}}{v_0*(1+(1-\frac{a_{drag\%}}{100})^t)}$$ But even after doing these simplifications of the formula you keep the base t and power t, which can't be solved using common logarithmic operations. I tried applying a lot of them as well as looking for other solutions like infinite powers but none seem to get me to a function where t isn't required to calculate itself. Which leads me to think that using limits is the way to solve this equation, because from programming experience I know that repeating this calculation over and over will in the end converge to a certain value since there is a time this object takes to travel to the destination.
If you are wondering how I made this equation, its made to function within a game to predict the arc of a projectile as accurate as possible. Since the games physics are different from real life (like drag being a percentage of speed per second) I ended up with this formula. In the end I would want to know the angle to shoot at, but that isn't relevant for this question since I could simplify the equation to not involve the angle anymore.
So now the real question: How would I, given this formula, make a function for t and thus get rid of the power in $(1-\frac{a_{drag\%}}{100})^t$?
I ploted the graph in desmos to verify that it actually converged and got the following results