Question

Find the value of x if
${4}^{2x - 1} - {16}^{x - 1} = 384$

Answer 1

${4 }^{2x - 1} - {16}^{x - 1} = 384$
we know that,

16 = (4)²

=>
${4}^{2x - 1} - ( {4}^{2} ) {}^{x - 1} = 384 \\ = > \frac{ {4}^{2x} }{4} - {4}^{2x - 2} = 384$
since,
${a}^{b - c} = \frac{ {a}^{b} }{ {a}^{c} }$
so,
$\frac{ {4}^{2x} }{4} - \frac{ {4}^{2x} }{ {4}^{2} } = 384 \\ = > {4}^{2x} ( \frac{1}{4} - \frac{1}{16} ) = 384 \\ \\ = > {4}^{2x} ( \frac{4 - 1}{16} ) = 384 \\ = > 4 {}^{2x} ( \frac{3}{16} ) = 384 \\ = > {4}^{2x} = 384 \times \frac{16}{3} \\ = > {4}^{2x} = 128 \times 16 \\ = > {4}^{2x} = 2048$
now we know that 4 = 2²

=>
${4}^{2x} = ( {2}^{2} ) {}^{2x}$
we have find that
${4}^{2x} = 2048$
also
$2048 = {2}^{11} \\ and \\ {4}^{2x} = ( {2}^{2} ) {}^{2x = {2}^{4x} } \\ = > {2}^{4x} = {2}^{11} \\ = > 4x = 11 \\ = > x = \frac{11}{4}$
hence, x = 11/4


Hope it helps dear friend ☺️✌️✌️

Answer 2

Answer :

Now,

4²ˣ⁻¹ - 16ˣ⁻¹ = 384

or, 4²ˣ⁻¹ - 4²ˣ⁻² = 384

or, 4²ˣ (4⁻¹ - 4⁻²) = 384

or, 4²ˣ (1/4 - 1/16) = 384

or, 4²ˣ {(4 - 1)/16} = 384

or, 4²ˣ (3/16) = 384

or, 4²ˣ = (384 × 16)/3

or, 4²ˣ = 2048

or, 4²ˣ = 2¹¹

or, 2⁴ˣ = 2¹¹

∴ comparing powers, we get

4x = 11

i. e., x = 11/4

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