Showing an inequality with ln I have to show that the following inequation is true: (ln(x) + ln(y))/(2) <= ln((x+y)/(2))

Madilyn Quinn 2022-10-23 Answered
Showing an inequality with ln
I have to show that the following inequation is true:
ln ( x ) + ln ( y ) 2 ln ( x + y 2 )
I transformed it into
ln ( x y ) 2 ln ( x + y ) ln ( 2 )
because I thought that I better can show the inequation here, but I don't know how to proceed.
How can I proceed or am I completely wrong?
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Answers (1)

Tirioliwo
Answered 2022-10-24 Author has 12 answers
How about we exponentiate both sides? We get
e ( l n ( x ) + l n ( y ) ) / 2 = e l n ( x ) / 2 e l n ( y ) / 2 = x y
and
e l n ( ( x + y ) / 2 ) = ( x + y ) / 2
Now the result is immediate per the AM-GM inequality.
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