A) AP
B) GP
C) HP
D) None of these
Correct Answer: C
Solution :
a, b, c are in GP. \[\therefore \] \[{{b}^{2}}=ac\] \[\Rightarrow \] \[2\log b=\log (ac)\] \[\Rightarrow \] \[2\log b=\log a+\log c\] \[\therefore \] \[\log a,\log b,\log c\]are in AP. \[{{\log }_{a}}x,{{\log }_{b}}x,{{\log }_{c}}x=\frac{\log x}{\log a},\frac{\log x}{\log b},\frac{\log x}{\log c}\] \[\therefore \] \[\frac{\log x}{\log a},\frac{\log x}{\log b},\frac{\log x}{\log c}\]are in AP. \[[\because \log a,\log b,\log c\,are\,in\,AP]\] Hence,\[{{\log }_{a}}x,{{\log }_{b}}x,{{\log }_{c}}x\]are in HP.
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