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, $$f(x) =log_{7}({ log_{3}(log_{5}(20x-x^{2}-91 )))}$$ => $$log_{3}(log_{5}(20x - x^{2} - 91 )) > 0$$ => $$log_{5}(20x - x^{2} - 91 ) > (3)^0$$ => $$log_{5}(20x - x^{2} - 91 ) > 1$$ => $$20x - x^{2} - 91 > (5)^1$$ => $$x^2 - 20x + 96 < 0$$ => $$(x - 8) (x - 12) < 0$$ => $$8 < x < 12$$ $$ herefore$$ Domain of $$f(x)$$ = $$(8,12)$$