\( \Large log \left(\frac{a+b}{2}\right) = \frac{1}{2} \left(log_{e}a + log_{e}b\right) \)

=> \( \Large \frac{a+b}{2} = \sqrt{ab} \)

=> \( \Large a+b-2\sqrt{ab} = 0 \)

=> \( \Large \sqrt{a} = \sqrt{b} => a = b \)