Aspire Faculty ID #10202 · Topic: NIMCET 2015 · Just now
NIMCET 2015

If a, b, c are in geometric progression, then $log_{ax}^{a}, log_{bx}^{a}$ and $log_{cx}^{a}$ are in

Solution

$ a, b, c $ are in G.P. 

$ \Rightarrow b^2 = ac $ 

Take logs base $a$: 

$ \log_a a = 1,\quad \log_a b,\quad \log_a c $ 

Since $ a, b, c $ are in G.P. 

$ \Rightarrow \log_a a,\ \log_a b,\ \log_a c $ are in A.P. 

Now given: 
$ \log_a(ax) = \log_a a + \log_a x = 1 + \log_a x $

$ \log_a(bx) = \log_a b + \log_a x $
 
$ \log_a(cx) = \log_a c + \log_a x $
 
These are: $ (1 + k),\ (\log_a b + k),\ (\log_a c + k) $ where $k = \log_a x$ 

Adding same constant does not change A.P. nature $\boxed{\text{They are in A.P.}}$
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