Strictly speaking an equation does not have a domain. A function does. Though it may make sense to speak about a solution set to this equation over a domain.
The following could be helpful towards an answer: The equation has two sides. The left hand side is a non-linear function and the right hand side is a line with slope -1 and y-intercept of 1.
The function[tex]\log_5(6-5^x)[/tex] has a domain over all Reals x for which it holds [tex]6-5^x > 0 \\6 > 5^x\\\log_5 6 > x[/tex] so the domain of the nonlinear function is [tex]-\infty<x<\log_5 6[/tex].
The domain of the linear function 1-x is all Reals: [tex]-\infty<x<+\infty[/tex]
So, together you could say the equation's solutions are defined over the domain [tex]-\infty<x<\log_5 6[/tex].
Lmk if you have questions.
