Sequence And Series Question 588
Question: If $ x,\ y,\ z $ are in G.P. and $ a^{x}=b^{y}=c^{z} $ , then
[IIT 1966, 68]
Options:
A) $ {\log_{a}}c={\log_{b}}a $
B) $ {\log_{b}}a={\log_{c}}b $
C) $ {\log_{c}}b={\log_{a}}c $
D) None of these
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Answer:
Correct Answer: B
Solution:
$ x,\ y,\ z $ are in G.P., then $ y^{2}=x\ .\ z $ Now $ a^{x}=b^{y}=c^{z}=m $
$ \Rightarrow $ $ x{\log_{e}}a=y{\log_{e}}b=z{\log_{e}}c={\log_{e}}m $
$ \Rightarrow $ $ x={\log_{a}}m,\ y={\log_{b}}m,z={\log_{c}}m $ Again as $ x,\ y,\ z $ are in G.P., so $ {{(p-r)}^{2}}={{(p+r)}^{2}}-4pr=16K^{2}-16K^{2}=0 $
$ \Rightarrow $ $ \frac{{\log_{b}}m}{{\log_{a}}m}=\frac{{\log_{c}}m}{{\log_{b}}m} $
$ \Rightarrow $ $ {\log_{b}}a={\log_{c}}b $ .
BETA
