Cho $\alpha ={{\log }_{a}}x,\beta ={{\log }_{b}}x.$ Khi đó ${{\log }_{a{{b}^{2}}}}\left( {{x}^{3}} \right)$ bằng

Ta có: ${{\log }_{a{{b}^{2}}}}\left( {{x}^{3}} \right)=3{{\log }_{a{{b}^{2}}}}x=\dfrac{3}{{{\log }_{x}}\left( a{{b}^{2}} \right)}=\dfrac{3}{{{\log }_{x}}a+2{{\log }_{x}}b}$
$=\dfrac{3}{\dfrac{1}{{{\log }_{a}}x}+\dfrac{2}{{{\log }_{b}}x}}=\dfrac{3{{\log }_{a}}x.{{\log }_{b}}x}{2{{\log }_{a}}x+{{\log }_{b}}x}=\dfrac{2\alpha \beta }{2\alpha +\beta }.$