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

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

UPSEEUPSEE 2008

Solution:

$\log _{4}(x-1)=\log _{2}(x-3)$
$\Rightarrow \log _{4}(x-1)=2 \log _{4}(x-3)$
$\Rightarrow \log _{4}(x-1)=\log _{4}(x-3)^{2}$
$\Rightarrow x-1=x^{2}+9-6 x$
$\Rightarrow x^{2}-7 x+10=0$
$\Rightarrow (x-5)(x-2)=0$
$\Rightarrow x=5$ or $2 $
Hence, $x=5 \{\because x=2 $ makes $\log (x-3)$ undefined $\}$
$\therefore $ Number of solution is $1 .$