Let $a=3 \sqrt{2}$ and $b=\frac{1}{5^{1 / 6} \sqrt{6}}$. If $x, y \in \mathbb{R}$ are such that $\begin{gathered}3 x+2 y=\log _a(18)^{\frac{5}{4}} \2 x-y=\log _b(\sqrt{1080})\end{gathered}$ and then $4 x+5 y$ is equal to . ________.

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Aicharya · Accepted Answer

$\begin{aligned} & 3 x+2 y=\log _{3 \sqrt{2}}(3 \sqrt{2})^{\frac{5}{2}}=\frac{5}{2} \ & \Rightarrow \quad 6 x+4 y=5 \quad \ldots \ldots(1)\end{aligned}$ $2 x-y=\log _{\frac{1}{5^{1 / 6} \sqrt{6}}}\left(5^{\frac{1}{6}} \sqrt{6}ight)^3=-3$ $\begin{array}{ll}\Rightarrow & 2 x-y=-3 \quad.....(2)\ & 	ext { equation (1) }-(2) \ \Rightarrow \quad & 4 x+5 y=8\end{array}$

Let $a=3 \sqrt{2}$ and $b=\frac{1}{5^{1 / 6} \sqrt{6}}$. If $x, y \in \mathbb{R}$ are such that…

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