I've found this interesting equation on the web: $$p-1 = (1 - e^{\alpha-\beta t})^{t+1}$$
which has to be solved for t, considering that the parameters: $\alpha, \beta, p$ are defined correctly.
First thought was to use some logarithms: $$\ln(p-1)=(t+1)\cdot ln(1-e^{\alpha-\beta t})$$ $$t+1=\frac{ln(p-1)}{ln(1-e^{\alpha-\beta t})}$$
Or, alternatively, I have: $$t+1=log_{1-e^{\alpha-\beta t}}(p-1)$$
I also thought about some substitutions than can be made, or an analytical approach.
Any guidance will be greatly appreciated.