[tex]\log_{10}5 = a \\ \log_{10} 6 =b \\ \\ \\ \log_{3}2 = \dfrac{\log_{10} 2}{\log_{10}3} = \dfrac{\log_{10}\dfrac{10}{5}}{\log_{10}{\dfrac{5\cdot 6}{10}}} = \dfrac{\log_{10}10 - \log_{10}5}{\log_{10}(5\cdot 6) - \log_{10}10} =\\ \\ \\=\dfrac{1-a}{\log_{10} 5+\log_{10} 6-1}} = \boxed{\dfrac{1-a}{a+b-1}}[/tex]
