Question

Calculate
$7^{{\log }_{3}5}+3^{{\log }_{3}7}-5^{{\log }_{3}7}-7^{{\log }_{5}3}$

Answer 1

$\begin{array}{c}Letx=7^{{{\log }_3}^5}\\ taking\log bothside\\ \log X=\log 7^{\left({{\log }_3}^5\right)}\\ ={{\log }_3}^5\log 7\\ =\frac{\log 5\times \log 7}{\log 3}\\ similarily\\ \log Y=\frac{\log 7\times \log 3}{\log 3}\\ \log Z=\frac{\log 7.\log 5}{\log 3}\\ \log A=\frac{\log 3.\log 7}{\log 7}\\ A/ctoquestion\\ \log X+\log Y-\log Z-\log A=\\ \frac{\log 5.\log 7}{\log 3}+\frac{\log 7.\log 3}{\log 5}-\frac{\log 5\times \log 7}{\log 3}=\frac{\log 7\times \log 3}{\log 5}\\ \log \left(\frac{XY}{ZA}\right)=0\\ \frac{XY}{ZA}=1\\ XY=ZA\end{array}$