Question

Whats the domain for this equation?

Answer 1

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
`\log_5(6-5^x)` has a domain over all Reals x for which it holds
`6-5^x > 0 \\6 > 5^x\\\log_5 6 > x` so the domain of the nonlinear function is
`-\infty<x<\log_5 6`.

The domain of the linear function 1-x is all Reals:
`-\infty<x<+\infty`

So, together you could say the equation's solutions are defined over the domain
`-\infty<x<\log_5 6`.

Lmk if you have questions.