if-a-2-b-2-3ab-then-prove-that-log-a-b-5-1-2-loga-logb-

Question Number 10064 by PradipGos. last updated on 22/Jan/17

if a^{2} + b^{2} = 3 ab then prove that

\log \frac{a + b}{\sqrt{5}} = \frac{1}{2} (\log a + \log b)

Answered by ridwan balatif last updated on 22/Jan/17

a^{2} + b^{2} = 3 ab \to {(a + b)}^{2} - 2 ab = 3 ab \to a + b = \pm \sqrt{5 ab}

we take a + b = + \sqrt{5 ab}

\log \frac{a + b}{\sqrt{5}} = \log \frac{\sqrt{5} \times \sqrt{ab}}{\sqrt{5}}

= \log \sqrt{ab}

= \log {(ab)}^{\frac{1}{2}}

= \frac{1}{2} \log (ab)

= \frac{1}{2} (loga + logb) (Terbukti)