Respuesta :

kudzordzifrancis
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]

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