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Basic of Logarithms
medium
Let $a=3 \sqrt{2}$ and $b=\frac{1}{5^{\frac{1}{6}} \sqrt{6}}$. If $x, y \in R$ are such that $3 x+2 y=\log _a(18)^{\frac{5}{4}} \text { and }$ $2 x-y=\log _b(\sqrt{1080}),$ then $4 x+5 y$ is equal to. . . .
A
$3$
B
$4$
C
$8$
D
$9$
(IIT-2024)
Solution
$3 x+2 y=\log _{3 \sqrt{2}}(3 \sqrt{2})^{\frac{5}{2}}=\frac{5}{2}$
$\Rightarrow \quad 6 x+4 y=5$ $. . . (1)$
$2 x – y =\log _{\frac{1}{5^{1 / 6} \sqrt{6}}}\left(5^{\frac{1}{6}} \sqrt{6}\right)^3=-3$
$\Rightarrow \quad 2 x – y =-3$ $. . . (2)$
$\text { equation (1) – (2) }$
$\Rightarrow \quad 4 x+5 y=8$
Standard 11
Mathematics
