Question

Find $lo{g}_{24}$ 48 in terms of $\alpha$ if $lo{g}_{12}$ 36 = $\alpha$.

Answer 1

${\log }_{24}48=$${\log }_{24}24+{\log }_{24}2=$$1+\frac{1}{{\log }_{2}24}$

${\log }_{2}24={\log }_{2}8+{\log }_{2}3=3+{\log }_{2}3$

$\alpha ={\log }_{12}36=2{\log }_{12}6$

$\Rightarrow \frac{\alpha }{2}=\frac{1}{{\log }_{6}12}$

${\log }_{6}12={\log }_{6}6+{\log }_{6}2=1+\frac{1}{{\log }_{2}6}=1+\frac{1}{1+{\log }_{2}3}$

Hence

$\frac{{\log }_{2}3+2}{1+{\log }_{2}3}=\frac{2}{\alpha }$

$\Rightarrow {\log }_{2}3=\frac{2(\alpha -1)}{2-\alpha }$

${\log }_{24}48=1+\frac{2-\alpha }{6-3\alpha +2\alpha -2}$

$=\frac{2(3-\alpha )}{4-\alpha }$