Answer:
[tex]\log (\frac{10x^{\frac{1}{2}}}{\sin ^22x})[/tex]Explanation:
Given the logarithmic expression:
[tex]\frac{1}{2}\log x-2\log (\sin 2x)+1[/tex]Using the laws of logarithm, the expression can be expressed as:
[tex]\begin{gathered} \log x^{\frac{1}{2}}-log(\sin 2x)^2+\log 10 \\ \end{gathered}[/tex]Since subtraction becomes division and addition become multiplication in logarithmic laws, the expression becomes:
[tex]\begin{gathered} =\log x^{\frac{1}{2}}+\log 10-\log (\sin 2x)^2 \\ =\log \sqrt[]{x}+\log 10-\log (\sin 2x)^2 \\ =\log (10\sqrt[]{x})-\log (\sin 2x)^2 \\ =\log (\frac{10x^{\frac{1}{2}}}{\sin ^22x}) \end{gathered}[/tex]This gives the required expression as a single logarithm
