ANSWER
The true solution is
[tex]x = \frac{4}{9} [/tex]
EXPLANATION
The logarithmic equation given to us is
[tex] log_{2}(6x) - log_{2}( \sqrt{x} ) = 2 [/tex]
We need to use the quotient rule of logarithms.
[tex] log_{a}( M ) - log_{a}( N ) = log_{a}( \frac{ M }{ N } ) [/tex]
When we apply this law the expression becomes
[tex] log_{2}( \frac{6x}{ \sqrt{x} } ) = 2 [/tex]
We now take the antilogarithm of both sides to get
[tex] \frac{6x}{ \sqrt{x} } = {2}^{2} [/tex]
[tex] \frac{6x}{ \sqrt{x} } = 4[/tex]
We square both sides to get,
[tex] (\frac{6x}{ \sqrt{x} }) ^{2} = {4}^{2} [/tex]
We evaluate to obtain,
[tex] \frac{36 {x}^{2} }{ x } = 16 [/tex]
This simplifies to
[tex]36x = 16[/tex]
We divide both sides by 36 to get
[tex]x = \frac{16}{36} [/tex]
We simplify to get,
[tex]x = \frac{4}{9} [/tex]
