If a = b^2 = c^3 = d^4, then the value of _a ( abcd ) would be -

Answer: Option C

$$\begin{array}{rl}& a=b^2=c^3=d^4\\ & \Rightarrow b=a^{\frac{1}{2}},c=a^{\frac{1}{3}},d=a^{\frac{1}{4}}\\ & \therefore {\log }_a\left(abcd\right)\\ & ={\log }_a\left(a\times a^{\frac{1}{2}}\times a^{\frac{1}{3}}\times a^{\frac{1}{4}}\right)\\ & ={\log }_a{a}^{\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right)}\\ & =\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right){\log }_{a}a\\ & =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\end{array}$$