[tex]\log_{\big2}8\sqrt2-\log_{\big3}3\sqrt3= \\ \\ = \log_{\big2}8+\log_{\big2}\sqrt2- \big(\log_{\big3}3+\log_{\big3} \sqrt3\big) = \\ \\ = 3+\log_{\big2}2^{\dfrac{1}{2}}-\Big(1+\log_{\big3}3^{\dfrac{1}{2}}\Big) = \\ \\ = 3+\dfrac{1}{2}\cdot \log_{\big2}2-\Big(1+\dfrac{1}{2}\cdot \log_{\big3}3\Big) = \\ \\ = 3+\dfrac{1}{2}-\Big(1+\dfrac{1}{2}\Big) = \\ \\ = \dfrac{7}{2} - \dfrac{3}{2}} = \\ \\ = \dfrac{4}{2} = \\ \\ =2[/tex]