Answer
(a) Domain: {$x∈R| x\ne0; x\ne1$}
(b) $f^{-1}(x)=10^{2^x}$
Work Step by Step
Quick review before we begin. $\log_x(a)$=b; $a\ne0$
$a=x^b$
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(a) So the domain of $f(x)=\log_2({\log_{10}{x}})$ is any number except that:
$\left\{ \begin{array}{ll} x\ne0\\ \log_{10}{x}\ne0 \end{array} \right. $ $\left\{ \begin{array}{ll} x\ne0\\ x\ne1 \end{array} \right. $
So, we have the Domain: {$x∈R| x\ne0; x\ne1$}
(b) We have: $y=\log_2({\log_{10}{x}})$. To find inverse of this function we have to switch $x$ and $y$:
$\log_2({\log_{10}{y}})=x$
${\log_{10}{y}}=2^x$
$y=10^{2^x}$
$f^{-1}(x)=10^{2^x}$
