Question

If $x=\log 2$ and $y=\log 3$ express ${\log }_{12}18$ & ${\log }_{24}54$ in terms $xandy$

Answer 1

Let, $\log 2=x$

$\log 3=y$

${\log }_{12}18=\frac{\log 18}{\log 12}=\frac{\log 3^2.2}{\log 2^2.3}=\frac{2\log 3+\log 2}{2\log 2+\log 3}$

$=\frac{x+2y}{2x+y}$

${\log }_{24}54=\frac{\log 54}{\log 24}=\frac{\log 3^3.2}{\log 2^3.3}$

$=\frac{3{\log }_3+\log 2}{3\log 2+\log 3}=\frac{x+3y}{3x+y}$