In English that would be "monotonic". What do you mean by "all of the functions are monotonic? And my problem is not with your saying that [tex]5^x + 7^x=12^x[/tex] is monotonic but that it is a function at all! That is an equation. Perhaps you are confusing "function" with "equation" but in that case, I don't recognise the phrase "monotonic equation".
In any case, the definition of "monotonic" is that if f(x)= f(y) then x= y. You cannot have two different values of the independent variable giving the same value of the dependent variable.
That means, in particular, that if f(x) is a monotonic function then the equation f(x)= a, for any a, cannot have more than one solution- it has either no solutions or one.
There is no general method for solving equations involving monotonic functions. Many polynomial functions, of odd degree, are monontonic but there is no general method of solving them.