Question:
If $\left(\frac{2}{3}\right)^{x}\left(\frac{3}{2}\right)^{2 x}=\frac{81}{16}$ then $x=?$
(a) 1
(b) 2
(c) 3
(d) 4
Solution:
$\left(\frac{2}{3}\right)^{x}\left(\frac{3}{2}\right)^{2 x}=\frac{81}{16}$
$\Rightarrow\left(\frac{2}{3}\right)^{x}\left(\frac{2}{3}\right)^{-2 x}=\frac{81}{16}$
$\Rightarrow\left(\frac{2}{3}\right)^{x-2 x}=\frac{81}{16}$
$\Rightarrow\left(\frac{2}{3}\right)^{-x}=\frac{3^{4}}{2^{4}}$
$\Rightarrow\left(\frac{3}{2}\right)^{x}=\frac{3^{4}}{2^{4}}$
$\Rightarrow x=4$
Hence, the correct answer is option (d).
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