From a practical perspective, my question can be most easily considered as solving for time in a future-value type equation, but for two separate investments growing at different rates. Say you have two savings accounts, one presently valued at $P_1$ and growing at a rate of $R_1$ (compounding annually), the other at $P_2$ growing at $R_2$. Starting from the standard compounding future-value formula: \begin{equation} F=P(1+R)^t \end{equation} you want to know how many years ($t$) will pass before their combined future value crosses a given threshold $X$ (i.e., $X=F_1+F_2$): \begin{equation} X = P_1(1+R_1)^{t} + P_2(1+R_2)^{t} \end{equation} Unfortunately, that is as far as I get. How do you solve for $t$?
Note: in this case, I am not interested in answers directed to plotting the curve or knowing the incidental value of $t$ for a specific set of inputs, rather, I want to know how to recast the equation in the form of $t = ?$.
Thanks.
Basically, you won't get an algebraic answer. You will need to do a numerical root finding approach. A nice upper bound is available by lowering the higher R to match the other, then back down to find the correct $t$