If (a+b 2 )=1 2 ( a+ b), then show that a=b. - Vidyadip

Question

If $\log \left(\frac{a+b}{2}\right)=\frac{1}{2}(\log a+\log b)$, then show that $a=b$.

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Answer

$ \log \left(\frac{a+b}{2}\right)=\frac{1}{2}(\log a+\log b)$
$\therefore 2 \log \left(\frac{a+b}{2}\right)=\log a+\log b$
$\therefore \log \left(\frac{a+b}{2}\right)^2=\log a b$
$\therefore \frac{(a+b)^2}{4}=a b$
$\therefore a^2+2 a b+b^2=4 a b$
$\therefore a^2+2 a b-4 a b+b^2=0$
$\therefore a^2-2 a b+b^2=0$
$\therefore(a-b)^2=0$
$\therefore a-b=0$
$\therefore a=b$

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