The number of solutions of log4(x - 1) = log2(x - 3) is

Question

The number of solutions of $\log_{4} (x - 1) = \log_{2} (x - 3)$ is

Smart Achievers · Accepted Answer

(2)

$For the given equation to be valid, we must have$ $x - 1 > 0 and x - 3 > 0$

$\Rightarrow$ $x > 3$

$We can write the given equation as \frac{\log (x - 1)}{\log 4} = \frac{\log (x - 3)}{\log 2}$

$\Rightarrow$ $\log (x - 1) = 2 \log (x - 3)$ $(∵ \log 4 = 2 \log 2)$

$\Rightarrow$ $(x - 1) = {(x - 3)}^{2} \Rightarrow x^{2} - 7 x + 10 = 0$

$\Rightarrow$ $(x - 5) (x - 2) = 0 \Rightarrow x = 5 or 2$

$As x > 3,$ $x = 5$

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