Let and . If are such that and , then is equal to ______. [2024]

Question

Let $a = 3 \sqrt{2}$ and $b = \frac{1}{5^{1 / 16} \sqrt{6}}$ . If $x, y \in ℝ$ 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 ______. [2024]

Smart Achievers · Accepted Answer

(8)

$a = 3 \sqrt{2} \Rightarrow a^{2} = 18; Now, 5^{\frac{1}{6}} \sqrt{6} = {(5 . 6^{3})}^{\frac{1}{6}} = {(1080)}^{\frac{1}{6}}$

$2 x - y = \log_{b} (\sqrt{1080})$ $b = \frac{1}{{(1080)}^{\frac{1}{6}}} = (1080) = b^{- \frac{1}{6}}$

$\Rightarrow 3 x + 2 y = \log_{a} {(a^{2})}^{\frac{5}{4}} = \frac{5}{2} ...(i)$

$2 x - y = \log_{a} b^{- 3} = - 3 ...(ii)$

$On solving (i) & (ii) we get,$

$x = \frac{- 1}{2}, y = 2 \Rightarrow 4 x + 5 y = 8$

Solution

Q.
Let $a = 3 \sqrt{2}$ and $b = \frac{1}{5^{1 / 16} \sqrt{6}}$ . If $x, y \in ℝ$ 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 ______. [2024]

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Solution

Q. Let a=32 and b=151/166. If x,y∈ℝ are such that 3x+2y=loga(18)54 and 2x-y=logb(1080), then 4x+5y is equal to ______. [2024]

Ans. (8) a=32⇒a2=18; Now, 5166=(5.63)16=(1080)16 ∴ b=1(1080)16=(1080)=b-16 ⇒3x+2y=loga(a2)54=52 ...(i) 2x-y=logab-3=-3 ...(ii) On solving (i) & (ii) we get, x=-12, y=2⇒4x+5y=8

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Q.
Let $a = 3 \sqrt{2}$ and $b = \frac{1}{5^{1 / 16} \sqrt{6}}$ . If $x, y \in ℝ$ 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 ______. [2024]