# Solving for exponent with multiple bases

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.

• For some basic information about writing math at this site see e.g. here, here, here and here. – user93957 Dec 20 '13 at 17:18
• "Basic information?" I wish, Adobe; all four of those sites are written for folks who already have some familiarity with how to do it, and only need some quick pointers for less common things -- not for a noob. I had to find and then read a good bit of Oetiker's "Not So Short Introduction to LATEX2e" to get my equations to work. I also noticed that other commenters at those four sites observed that figuring out how to write formulae here is neither obvious nor easy. – Bruce Dec 23 '13 at 21:42

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 $t$