Can I show that the following expression is always positive,

b−a+a(loga−logb)b-a+a(\log a – \log b)

here a>ba>b and both aa and bb are positive numbers between 0 and 1. I have plotted this expression and I can see that it is positive but I want to show it analytically. Can anyone help ?

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maybe it can be considered a mathematica problem

– Wjx

Sep 8 at 15:02

related: mathematica.stackexchange.com/q/58181/9490

– JasonB

Sep 8 at 15:14

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1 Answer

1

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Divide by b (remember b > 0) and substitute x = a/b to get

1 – x + x Log[x] > 0

Then

Minimize[1 + x Log[x] – x, x]

(* {0, {x -> 1}} *)

Or analytically:

δδx(1−x+xlogx)=−1+logx+xx=logx\frac{\delta}{\delta x}\left(1-x+x\log{x}\right)=-1+\log{x}+\frac{x}{x}=\log{x}

Which is 00 when x=1x=1.

f(x=1)=1−1+1log1=0f\left(x=1\right)=1-1+1\log{1}=0

But x <= 1 is not allowed because a > b. The expression is positive for x > 1.