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Q.
The product of all the solutions of the equation $\log _x(3) \cdot \log _{\frac{x}{3}}(3)+\log _{\frac{x}{81}}(3)=0$ is
Continuity and Differentiability
Solution:
$ \frac{1}{\log _3 x } \cdot \frac{1}{\left(\log _3 x -1\right)}+\frac{1}{\left(\log _3 x -4\right)}=0$
$\text { Put } \log _3 x = t $
$ ( t -4)+ t ( t -1)=0$
$ t ^2=4 \Rightarrow t = \pm 2 $
$ \log _3 x =2 \Rightarrow x =9 $
$\log _3 x =-2 \Rightarrow x =\frac{1}{9}$