Question

log3 (x - 2) - log9 (x - 2) = 2

Answer 1

$log_3(x-2)-log_9(x-2)=2 \\ \\ log_3(x-2)- \frac{log_3(x-2)}{log_39} =2 \\ \\ log_3(x-2)-\frac{log_3(x-2)}{2}=2 \\ \\ 2log_3(x-2)-log_3(x-2)=4 \\ \\ log_3(x-2)=4 \\ \\ x-2=3^4 \\ \\ x-2=81 \\ \\x=81+2 \\ \\ x=83 \\ \\ \\ S=\{83\}$

Answer 2

$\log_3(x-2)-\log_9(x-2)=2\\\\\log_3(x-2)-\log_{3^2}(x-2)=2\\\\\log_3(x-2)-\frac{1}{2}\cdot\log_3(x-2)=2\\\\\frac{\log_3(x-2)}{1/2}-\frac{\log_3(x-2)}{2/1}=\frac{2}{1/2}\\\\2\cdot\log_3(x-2)-\log_3(x-2)=4\\\\\log_3(x-2)=4\\\\3^4=x-2\\\\x-2=81\\\\\boxed{x=83}$