If $$a = {b^2} = {c^3} = {d^4},$$ then the value of $${\log _a}\left( {abcd} \right)$$ would be - Options: A . &nbsp$${\log _a}1 + {\log _a}2 + {\log _a}3 + {\log _a}4$$ B . &nbsp$${\log _a}24$$ C . &nbsp$${\text{1 + }}\frac{1}{2} + \frac{1}{3} + \frac{1}{4}$$ D . &nbsp$${\text{1 + }}\frac{1}{{2!}} + \frac{1}{{3!}} + \frac{1}{{4!}}$$

If $$a = {b^2} = {c^3} = {d^4},$$ then the value of $${\log

Question

If $$a = {b^2} = {c^3} = {d^4},$$ then the value of $${\log _a}\left( {abcd} \right)$$ would be -

Options:

A . $${\log _a}1 + {\log _a}2 + {\log _a}3 + {\log _a}4$$

B . $${\log _a}24$$

C . $${\text{1 + }}\frac{1}{2} + \frac{1}{3} + \frac{1}{4}$$

D . $${\text{1 + }}\frac{1}{{2!}} + \frac{1}{{3!}} + \frac{1}{{4!}}$$

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

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