Question 848902: Given that a,b,c, form a geometric sequence,show that loga:logb:logc form an arthimetic sequence
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! log a, log b, log c are in arithmetic progression iff 2 log b = log a + log c <--> log (b^2) = log ac (by logarithmic properties) <--> b^2 = ac, since the log function is one-to-one.
Note that b^2 = ac holds iff a/b = b/c, which is the criterion for a geometric sequence. We are assuming all terms are positive (although the statement should hold if a,b,c are negative, except log a, etc. is complex).
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