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Question
If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
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Solution
a, b, c are in G.P.
\[\therefore b^2 = ac \]
\[\text { Now taking } lo g_m \text { on both the sides: } \]
\[ \Rightarrow lo g_m \left( b \right)^2 = lo g_m \left( ac \right)\]
\[ \Rightarrow 2lo g_m \left( b \right) = lo g_m a + lo g_m \left( c \right)\]
\[ \Rightarrow \frac{2}{\log_b \left( m \right)} = \frac{1}{\log_a \left( m \right)} + \frac{1}{\log_c \left( m \right)}\]
\[\text { Thus }, \frac{1}{\log_a \left( m \right)}, \frac{1}{\log_b \left( m \right)} \text { and } \frac{1}{\log_c \left( m \right)} \text { are in A . P } . \]
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