Find the value of $x$ which satisfies the equation : ${log}_{2} (x - 16) = {log}_{4} (x - 4)$

12

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13

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16

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20

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24

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Solution

The correct option is D $20$
${log}_{2} (x - 16) = {log}_{4} (x - 4) = {log}_{2^{2}} (x - 4) = \frac{1}{2} {log}_{2} (x - 4) = {log}_{2} \sqrt{x - 4}$
Now we can compare powers , we get $x - 16 = \sqrt{x - 4}$ , square it on both sides , we get
$x - 4 = {(x - 16)}^{2} = x^{2} - 32 x + 256$
$x^{2} - 33 x + 260 = 0$
Therefore $x = 20$ satisfies this equation

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