Context: I want to find the amount of time it takes for the sum of exponential growth to reach a given value.
$x_t=\sum\limits_{i=0}^{t}x_0(1+r)^i$
In other words, is there a formula I can use to solve for $t$?
If there is a solution, can you please also show the derivation?
An obvious Ansatz is $x_t=A(1+r)^t+B$ for constants $A,\,B$ you can find from the cases $t=0,\,t=1$. You'll get$$x_t=\frac{x_0}{r}[(1+r)^{t+1}-1],$$which can also prove by induction on $t$. This equation is obviously correct when $t=0$, and if it works with $t=k$ then$$\begin{align}x_{k+1}&=\frac{x_0}{r}[(1+r)^{k+1}-1]+x_0(1+r)^{k+1}\\&=\frac{x_0}{r}[(1+r)^{k+1}(1+r)-1]\\&=\frac{x_0}{r}[(1+r)^{k+1+1}-1].\end{align}$$(@JohnWaylandBales has also noted a telescoping sum gets this result.) Solving for $t$ gives $$(1+r)^{t+1}=1+\frac{rx_t}{x_0}\implies t=\frac{\ln\left(1+\frac{rx_t}{x_0}\right)}{\ln(1+r)}-1.$$