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Basic of Logarithms
medium
If ${\log _e}\left( {{{a + b} \over 2}} \right) = {1 \over 2}({\log _e}a + {\log _e}b)$, then relation between $a$ and $b$ will be
A
$a = b$
B
$a = {b \over 2}$
C
$a =2 b$
D
$a = {b \over 3}$
Solution
(a) ${\log _e}\left( {{{a + b} \over 2}} \right) = {1 \over 2}({\log _e}a + {\log _e}b)$
$ = {1 \over 2}{\log _e}(ab) = {\log _e}\sqrt {ab} $
$ \Rightarrow {{a + b} \over 2} = \sqrt {ab} \,\, \Rightarrow a + b = 2\sqrt {ab} $
==> $\,a = b$.
Standard 11
Mathematics
