Question

Find ${\log }_{25}24$ if ${\log }_{6}15=\alpha$ and ${\log }_{12}18=\beta$ .

Answer 1

$\begin{array}{c}{\log }_6^{15}=\alpha ,{\log }_{12}^{18}=\beta ,\\ let{\log }_{25}^{24}=c\\ 2c=\frac{\left(5-\beta \right)}{\left(\alpha \beta +\alpha -2\beta +1\right)}\\ 2c=\frac{\left(5-{\log }_{12}^{18}\right)}{\left({\log }_6^{15}.{\log }_{12}^{18}+{\log }_6^{15}-2{\log }_{12}^{18}+1\right)}\\ \therefore c={\log }_{25}^{24}=\frac{\left(5-{\log }_{12}^{18}\right)}{2\left\{{\log }_6^{15}\left({\log }_{12}^{18}+1\right)-2{\log }_{12}^{18}+1\right\}}\end{array}$