Explanation

(a)
$\log_{3}{x + \log_{3}{\sqrt{x} + \log_{3}{\sqrt[4]{x} + \log_{3}{\sqrt[8]{x} + ... = 4}}}}$
$\Rightarrow \log_{3}{x + \frac{1}{2} + \log_{3}{x + \frac{1}{4}\log_{3}{x + \frac{1}{8}\log_{3}{x + ... = 4}}}}$
$\Rightarrow \log_{3}{x\left\lbrack 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... \right\rbrack} = 4$
$\Rightarrow \log_{3}{x\left\lbrack \frac{1}{1 - \frac{1}{2}} \right\rbrack = 4}$
$\Rightarrow \log_{3}{x = 2}$
$\Rightarrow x = 3^{2} = 9$