SOLUTION: For a= Sqrt(10)+ sqrt(2)/2 and b= Sqrt(10)- sqrt(2)/2 , we have Log2 (a^2+ab+b^2)= ?

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Question 878746: For a= Sqrt(10)+ sqrt(2)/2 and b= Sqrt(10)- sqrt(2)/2 , we have
Log2 (a^2+ab+b^2)= ?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Is that log%282%2C%28a%5E2%2Bab%2Bb%5E2%29%29 ? That looks like a lot of calculation work, no matter how creatively you try to do it.

DIVIDE AND CONQUER:
a%5E2%2Bab%2Bb%5E2=la%5E2%2B2ab%2Bb%5E2-ab=%28a%2Bb%29%5E2-ab

,
and %28a%2Bb%29%5E2=%282sqrt%2810%29%29%5E2=2%5E2%28sqrt%2810%29%29%5E2=4%2A10=40
So a%5E2%2Bab%2Bb%5E2=%28a%2Bb%29%5E2-ab=40-19%2F2=80%2F2-19%2F2=61%2F2
and
An approximate value can be calculated (by a calculator) or using a table of logaritms) as
log%282%2C61%29-1=log%2861%29%2Flog%282%29-1=approx1.78533%2F0.30103=about5.93-1=about4.93

GRIT YOUR TEETH AND GO:
log%282%2C%28a%5E2%2Bab%2Bb%5E2%29%29
=
=
=
=
=
=

A CREATIVE IDEA THAT DID NOT WORK TOO WELL:
a%5E2%2Bab%2Bb%5E2 gives me an idea:
a%5E2%2Bab%2Bb%5E2=%28a%5E3-b%5E3%29%2F%28a%2Bb%29

a%5E3-b%5E3=%28sqrt%2810%29%2Bsqrt%282%29%2F2%29%5E3-%28sqrt%2810%29-sqrt%282%29%2F2%29%5E3
=%22%5B%22%22%5D+-+%5B%22%22%5D%22
=%22%5B%2210sqrt%2810%29%2B3%2810%29%28sqrt%282%29%2F2%29%2B3sqrt%2810%29%281%2F2%29%2Bsqrt%282%29%2F4%22%5D+-+%5B%2210sqrt%2810%29-3%2810%29%28sqrt%282%29%2F2%29%2B3sqrt%2810%29%281%2F2%29-sqrt%282%29%2F4%22%5D%22
=
=
=30sqrt%282%29%2F2%2Bsqrt%282%29%2F4%2B30sqrt%282%29%2F2%2Bsqrt%282%29%2F4
=
So and