What are the number of solutions of the equation ${log}_{4} (x - 1) = {log}_{2} (x - 3)$

1

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2

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3

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4

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Solution

The correct option is A $1$
${log}_{4} (x - 1) = {log}_{2} (x - 3)$
$\frac{1}{2} {log}_{2} (x - 1) = {log}_{2} (x - 3)$
$(x - 1)^{\frac{1}{2}} = x - 3$
$\sqrt{x - 1} = x - 3$
Squaring both sides we get,
$x - 1 = x^{2} + 9 - 6 x$
Or, $x^{2} - 7 x + 10 = 0$
$(x - 5) (x - 2) = 0$
Hence, the values of $x = 5$
And they have 1 possible solution.

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