The domain of the function $$f(x) =log_{7}({ log_{3}(log_{5}(20x-x^{2}-91 )))}$$ is:

A logarithm function of the form $$log_a b$$ is true, iff $$b > 0$$

and $$c = a^b$$ can be written as = $$log_a c = b$$

We have, $$f(x) =log_{7}({ log_{3}(log_{5}(20x-x^{2}-91 )))}$$

=> $$log_{3}(log_{5}(20x - x^{2} - 91 )) > 0$$

=> $$log_{5}(20x - x^{2} - 91 ) > (3)^0$$

=> $$log_{5}(20x - x^{2} - 91 ) > 1$$

=> $$20x - x^{2} - 91 > (5)^1$$

=> $$x^2 - 20x + 96 < 0$$

=> $$(x - 8) (x - 12) < 0$$

=> $$8 < x < 12$$

$$\therefore$$ Domain of $$f(x)$$ = $$(8,12)$$