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  4. Suppose that x is real number such that ((27)(9x)/4x)=(3x/8x), then the value of (2)-(1+ log 2 3) x, is

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Q. Suppose that $x$ is real number such that $\frac{(27)\left(9^x\right)}{4^x}=\frac{3^x}{8^x}$, then the value of $(2)^{-\left(1+\log _2 3\right) x}$, is

Continuity and Differentiability

Solution:

$(27) \cdot \frac{\left(9^x\right)}{3^x}=\frac{4^x}{8^x} \Rightarrow 27 \cdot 3^x=\frac{1}{2^x} \Rightarrow 27=6^{-x}$ ....(1)
Now $(2)^{-\left(1+\log _2 3\right) x}=(2)^{-x} \cdot(2)^{-\log _2(3)^x}=\frac{2^{-x}}{2^{\log _2 3^x}}=\frac{2^{-x}}{3^x}=6^{-x}$ .....(2)
Hence $(2)^{-\left(1+\log _2 3\right) x }=27$