Question 606592
*[tex \LARGE y = \log_9 {\log_8 {\log_7 {\log_6{ \log_5{x}}}}}]


First, we should find where x is defined. Note that the domain of any log function is always the positive real numbers (unless we are dealing with complex logarithms -- which we are not).


Hence,


*[tex \LARGE \log_8 {\log_7 {\log_6{ \log_5{x}}}} > 0]


*[tex \LARGE \log_7 {\log_6{ \log_5{x}}} > 8^0 = 1]


*[tex \LARGE \log_6{ \log_5{x}} > 7^{8^0} = 7] (see the pattern?)


*[tex \LARGE \log_5{x} > 6^{7^{8^0}} = 6^7]


*[tex \LARGE x > 5^{ 6^{7^{8^0}}} = 5^{6^7}]


Therefore any value of x such that *[tex \LARGE x > 5^{6^7}] is in the domain. Since log x is a differentiable function, log(log(x)) is differentiable, and so on. The domain of dy/dx is the same as the domain of y.