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Question Number 138091 by liberty last updated on 10/Apr/21

$$If{3}^{{\left(\log {}_{3}7\right)}^x}=7^{{\left(\log {}_{7}3\right)}^x},thenthe$$$$valueofxwillbe\dots$$

Answered by mitica last updated on 10/Apr/21

$${\left({log}_{3}7\right)}^x=a\Rightarrow {\left({log}_{7}3\right)}^x=\frac{1}{a}$$$$3^a=7^{\frac{1}{a}}\Rightarrow a={log}_3{7}^{\frac{1}{a}}\Rightarrow a={\left({log}_{3}7\right)}^{\frac{1}{2}}\Rightarrow x=\frac{1}{2}$$$$$$

Answered by Ankushkumarparcha last updated on 10/Apr/21

$$Solution:Taking{}^{\prime }{\log }_e^{\prime }onbothsides.$$$${\left({\log }_{3}7\right)}^x{\log }_e\left(3\right)={\left({\log }_{7}3\right)}^x{\log }_e\left(7\right)$$$${\left(\frac{{\log }_{3}7}{{\log }_{7}3}\right)}^x=\frac{{\log }_{e}7}{{\log }_{e}3},$$$${\left(\frac{{\log }_{e}7}{{\log }_{e}3}\right)}^{2x}=\frac{{\log }_{e}7}{{\log }_{e}3}(\because {\log }_{a}b=\frac{{\log }_{e}b}{{\log }_{e}a})$$$$Bycomparing.Weget,$$$$x=1/2$$

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