Explanation

(a)
$\left\lbrack 4\left( 1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + ... \right) \right\rbrack^{\log_{2}x} = \left\lbrack 54\left( 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + ... \right) \right\rbrack^{\log_{x}2}$
$\Rightarrow \left\lbrack 4\left( \frac{1}{1 + 1/3} \right) \right\rbrack^{\log_{2}x} = \left\lbrack 54\left( \frac{1}{1 - 1/3} \right) \right\rbrack^{\log_{x}2}$
$\Rightarrow \left\lbrack 4\left( \frac{3}{4} \right) \right\rbrack^{\log_{2}x} = \left\lbrack 54 \times \frac{3}{2} \right\rbrack^{\log_{x}2}$
$\Rightarrow 3^{\log_{2}x} = 3^{4\log_{x}2}$
${\Rightarrow \log_{2}}{x = 4\log_{x}{2 = \frac{4}{\log_{2}x}}}$
${\Rightarrow \left( \log_{2}x \right)}^{2} = 4 \Rightarrow \log_{2}{x = \pm 2}$
If $\log_{2}x = 2$
$\Rightarrow x = 2^{2} = 4$
And if $\log_{2}{x = - 2}$
$\Rightarrow \ \ \ \ x = 2^{- 2} = \frac{1}{4}$
$\therefore$ Solution set of the equation is $\left\{ 4,\ \frac{1}{4} \right\}$