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Basic of Logarithms
normal
यदि ${\log _e}\left( {\frac{{a + b}}{2}} \right) = \frac{1}{2}({\log _e}a + {\log _e}b)$ हो, तो $a $ और $b$ के मध्य सम्बंध होगा
A
$a = b$
B
$a = \frac{b}{2}$
C
$2a = b$
D
$a = \frac{b}{3}$
Solution
${\log _e}\left( {\frac{{a + b}}{2}} \right) = \frac{1}{2}({\log _e}a + {\log _e}b)$$ = \frac{1}{2}{\log _e}(ab) = {\log _e}\sqrt {ab} $
$ \Rightarrow \frac{{a + b}}{2} = \sqrt {ab} \,\, \Rightarrow a + b = 2\sqrt {ab} $$ \Rightarrow {\left( {\sqrt a – \sqrt b } \right)^2} = 0\,\, \Rightarrow \sqrt a – \sqrt b = 0$$ \Rightarrow \,\,\,\,a = b$
Standard 11
Mathematics
