Explanation

(c)
$x^{\left( \log_{10}{x\ } \right)^{2} - 3\log_{10}{x + 1}} > 1000 = 10^{3}$
$\Rightarrow \lbrack\left( \log_{10}x \right)^{2} - 3\log_{10}{x + 1\rbrack\log_{10}{x > 3\log_{10}{10 = 3}}}$
$\Rightarrow \left( \log_{10}x \right)^{3} - 3\left( \log_{10}{x\ } \right)^{2} + \log_{10}{x > 3}$
$\Rightarrow (\log_{10}{x)({\log_{10}x}{- 3) + 1(\log_{10}{x - 3) > 0}}}$
$\Rightarrow (\log_{10}{x - 3)(\log_{10}{x + 1) > 0}}$
$\Rightarrow \log_{10}{x - 3 > 0 \Rightarrow \log_{10}{x > 3}}$
$\Rightarrow x > 10^{3} = 1000$
$\Rightarrow x \in (1000,\infty)$