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Answer:
[tex]x=\frac{1}{4}[/tex]
Step-by-step explanation:
Given:
[tex]2log(\frac{2}{3})=\frac{1}{2}log(x)-log(18)+log(16)[/tex]
Use the Power Rule Law:
[tex]log(\frac{2}{3}^2)=log(x^{1/2})+log(16)-log(18)[/tex]
Use the Quotient Rule Law:
[tex]log(\frac{4}{9})=log(\sqrt{x})+log(\frac{16}{18})[/tex]
Use the Product Rule Law:
[tex]log(\frac{4}{9})=log(\frac{16\sqrt{x}}{18})[/tex]
Simplify:
[tex]log(\frac{4}{9})=log(\frac{8\sqrt{x}}{9})[/tex]
[tex]\frac{4}{9}=\frac{8\sqrt{x}}{9}[/tex]
[tex]4=8\sqrt{x}[/tex]
[tex]\frac{4}{8}=\sqrt{x}[/tex]
[tex]\frac{1}{2}=\sqrt{x}[/tex]
[tex]\frac{1}{4}=x[/tex]