SOLUTION: If a^2+b^2 = 7ab, show that log a+b/3=1/2 (loga + logb)

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Question 87641: If a^2+b^2 = 7ab, show that log a+b/3=1/2 (loga + logb)
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
Starting with +log%28%28a%2Bb%29%2F3%29+=+%281%2F2%29%28loga+%2B+logb%29+ i shall re-write it as +2log%28%28a%2Bb%29%2F3%29+=+%28loga+%2B+logb%29+.
I am going to prove this is correct instead by looking at the Left hand side +2log%28%28a%2Bb%29%2F3%29+ and showing that it equals the right hand side +%28loga+%2B+logb%29+.

So,
+2log%28%28a%2Bb%29%2F3%29+
+log%28%28%28a%2Bb%29%2F3%29%29%5E2+ --> one of laws of Logs
+log%28%28a%5E2%2B2ab%2Bb%5E2%29%2F9%29+ --> multiply out brackets

now we are told that +a%5E2%2Bb%5E2+=+7ab+ so sub that into what we have and we get:
+log%28%28a%5E2%2Bb%5E2%2B2ab%29%2F9%29+ re-arrange order so you can see the squared terms together
+log%28%287ab%2B2ab%29%2F9%29+ --> sub in the other equality
+log%28%289ab%29%2F9%29+ --> add together like terms
+log%28ab%29+ --> cancel out the 9's
+log%28a%29+%2B+log%28b%29+ use one of the Laws of Logs

which is what i set out to prove it equalled.

cheers
jon