Question

Find the number of solution(s) of the equation
${\log }_9(x+1).{\log }_2(x+1)-{\log }_9(x+1)-{\log }_2(x+1)+1=0$

Answer 1

${\log }_9(x+1){\log }_2(x+1)-{\log }_9(x+1)-{\log }_2(x+1)+1=0$

$\Rightarrow {\log }_9(x+1)({\log }_2(x+1)-1)-1({\log }_2(x+1)-1)=0$

$\Rightarrow ({\log }_9(x+1)-1)({\log }_2(x+1)-1)=0$

$\Rightarrow ({\log }_9(x+1)-1)=0,({\log }_2(x+1)-1)=0$

$\Rightarrow {\log }_9(x+1)=1,{\log }_2(x+1)=1$

$\Rightarrow x+1=9,x+1=2$

$\Rightarrow x=9-1,x=2-1$

$\therefore x=8,1$

$\therefore$number of solutions$=2$