Let a = 3√2 and b = 1/5^(1/6)√6. If x,y ∈ R are such that

Question

Let a \(=3\sqrt {2}\) and b \(=\frac {1}{5 ^{\frac {1}{6}}\sqrt {6}}.\) if \(x, y \in \mathbb{R}\) are such that

\(3 x+2 y=\log _{a}(18)^{\frac{5}{4}}\)and

\(2 x-y=\log _{b}(\sqrt{1080}),\)

then \(4 x+5 y\) is equal to ____.

Answer 1

Correct answer is : 8

\(a=3 \sqrt{2} \Rightarrow a^{2}=18\)

Notice that \(1080=5 \cdot 6^{3} \)

\(\Rightarrow 5^{\frac{1}{6}} \cdot 6^{\frac{1}{2}}=(1080)^{\frac{1}{6}}=\frac{1}{b} \)

\(\Rightarrow 1080^{\frac{1}{2}}=\frac{1}{b^{3}} \)

\(\Rightarrow 3 x+2 y=\log _{a}\left(a^{2}\right)^{\frac{5}{4}}=\frac{5}{2} \)  ...(i)

\(2 x-y=\log _{b} \frac{1}{b^{3}}=\log _{b} b^{-3}=-3\) ...(ii)

⇒ Solving (i) & (ii)

\(\Rightarrow x=\frac{-1}{2}, y=2 \)

\(\Rightarrow 4 x+5 y=8\)