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Q.
If $x >1, y >1, z >1$ are in G.P., then $\log _{ ex } e , \log _{ cy } e , \log _{ cz } e$ are in
Sequences and Series
Solution:
If $x>1, y>1, z>1$ are in G.P.
$\therefore y ^2= xz$
$\therefore 2 \ln y =\ln x +\ln z $
$\therefore \ln x , \ln y , \ln z \text { are in A.P. } $
$\therefore 1+\ln x , 1+\ln y ; 1+\ln z \text { are in A.P. }$
$\therefore \frac{1}{1+\ln x}, \frac{1}{1+\ln y}, \frac{1}{1+\ln z}$ are in H.P. $\Rightarrow \frac{1}{\ln e x}, \frac{1}{\ln \text { ey }}, \frac{1}{\ln e z}$ are in H.P. or $ \log _{ ex } e , \log _{ cy } e , \log _{ ez } e$ are in H.P.