$f (x) = \log_{10} (\log_{10} (\log_{10} (\log_{10} (\log_{10} x))))$ is defined iff $x \in$

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(10, \infty)

(10^{10}, \infty)

(10^{10^{10}}, \infty)

(10^{10^{10^{10}}}, \infty)

answer is C.

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Detailed Solution

f (x) = \log_{10} (\log_{10} (\log_{10} (\log_{10} (\log_{10} x))))

is defined

$iff \log_{10} (\log_{10} (\log_{10} (\log_{10} (\log_{10} x)))) > 0$

$\log_{10} (\log_{10} (\log_{10} x)) > 10^{10}$

$\log_{10} (\log_{10} (\log_{10} x)) > 1$

$\log_{10} (\log_{10} x) > 10$

$\log_{10} x > 10^{10} \Rightarrow x > 10^{10^{10}}$

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