If $(4)^{2 x - 1} - (16)^{x - 1} = 384$ ,then find the value of x.

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Solution

$4^{(2 x - 1)} - 16^{(x - 1)} = 384 \frac{4^{2 x}}{4} - \frac{16^{x}}{16} = 384 \frac{4 (4^{2 x}) - 16^{x}}{16} = 384 \frac{16^{x} (4 - 1)}{16} = 384 \Rightarrow 16^{x} = \frac{384 \times 16}{3} = \frac{2^{7} \times 3 \times 2^{4}}{3}$
$\Rightarrow 2^{4 x} = 2^{1} 1$
$\Rightarrow 4 x = 11$
$\Rightarrow x = \frac{11}{4} = 2.75$

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