Any idea for solving an equation of the type: $(t^a+x)^{1/a}=(t^b+x)^{1/b}$?

Question

I am trying to solve this equation:

$$(t^{\beta_X}-log(1-p))^{1/\beta_X}=(t^{\beta_Y}-log(1-p))^{1/\beta_Y},$$

where $t,\beta_X,\beta_Y>0$, $p\in(0,1)$. I know that there is a solution under some conditions as you can see here:

http://shiny.dmat.ua.es:3838/apps/shinyest/weibull/

but... I am not able to obtain an analytical expression. If anyone could give me some idea, I would be very grateful.

Thank you in advance,

Julio