Given that $:a = b^{2} = c^{3} = d^{4}$
we can take $a = b^{2}\implies \log_{a}b = \dfrac{1}{2}$
Similarly, $a = c^{3}\implies \log_{a}c = \dfrac{1}{3}$
and , $a = d^{4}\implies \log_{a}d = \dfrac{1}{4}$
Now, $\log_{a}(abcd) = \log_{a}a +\log_{a}b +\log_{a}c +\log_{a}d $
$\implies \log_{a}(abcd) = 1 + \dfrac{1}{2}+ \dfrac{1}{3}+ \dfrac{1}{4}$
Reference:https://brilliant.org/wiki/logarithms/
So, the correct answer is $(C).$