Question

If (3^log√x-1 to base 3) < (3^log(x-6) to base 3) +3 , what is x?

Answer 1

Final Answer : x > 6 .

Steps :
1)

Domain of Given function :
$x > 1 \: \: and \: x > 6 - - - (1)$
2)
${3}^{ log_{3}( \sqrt{x - 1} ) } < {3}^{ log_{3}(x - 6) } + 3 \\ = > { \sqrt{x - 1} }^{ log_{3}(3) } < {(x - 6)}^{ log_{3}(3) } + 3 \\ = > \sqrt{x - 1} < ( x - 6) + 3\\ = > \sqrt{x - 1} < x - 3$
3) Since both sides are positive by domain constraint.
so squaring both sides .we get.
$= > x - 1 < {x}^{2} - 6x + 9 \\ = > {x}^{2} - 7x + 10 > 0 \\ = > (x - 5)(x - 2) > 0 \\ = > x < 2 \: \: \: \: or \: \: x \: >5$

4) By, intersection of domain with these solution,

x > 6