Answer & Solution
Answer: Option C
Solution:
$$\eqalign{
& a = {b^2} = {c^3} = {d^4} \cr
& \Rightarrow b = {a^{\frac{1}{2}}},\,\,\,\,c = {a^{\frac{1}{3}}},\,\,\,\,d = {a^{\frac{1}{4}}} \cr
& \therefore {\log _a}\left( {abcd} \right) \cr
& = {\log _a}\left( {a \times {a^{\frac{1}{2}}} \times {a^{\frac{1}{3}}} \times {a^{\frac{1}{4}}}} \right) \cr
& = {\log _a}{a^{\left( {1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \right)}} \cr
& = \left( {1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \right){\log _a}a \cr
& = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \cr} $$