SOLUTION: If a,b,c………… be in G.P,then show that loga x,logb x,logc x, ……… are in H.P

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Question 971266: If a,b,c………… be in G.P,then show that loga x,logb x,logc x, ……… are in H.P
Answer by anand429(138) About Me  (Show Source):
You can put this solution on YOUR website!
Given, a,b,c are in GP,
So, let common factor be k such that b=ak and c=ak^2.
Now for checking the H.P. of given terms,
1%2Flog%28a%2Cx%29-1%2Flog%28b%2Cx%29
=log%28x%2Ca%29-log%28x%2Cb%29
=log%28x%2C%28a%2Fb%29%29
=log%28x%2C%28a%2F%28a%2Ak%29%29%29
=log%28x%2C%281%2Fk%29%29 -(i)
Again,
1%2Flog%28b%2Cx%29-1%2Flog%28c%2Cx%29
=log%28x%2Cb%29-log%28x%2Cc%29
=log%28x%2C%28b%2Fc%29%29
=log%28x%2C%28%28a%2Ak%29%2F%28a%2Ak%5E2%29%29%29
=log%28x%2C%281%2Fk%29%29 -(ii)
So, from (i) and (ii),
1%2Flog%28a%2Cx%29-1%2Flog%28b%2Cx%29+=+1%2Flog%28b%2Cx%29-1%2Flog%28c%2Cx%29
Hence, log%28a%2Cx%29,log%28b%2Cx%29 and log%28c%2Cx%29 are in H.P.