solve this

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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