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        riddles >> medium >> geometric progression
        
(Message started by: Christine on Apr 12^th, 2013, 9:28am)

Title: geometric progression
Post by Christine on Apr 12^th, 2013, 9:28am

For what number pairs x, y will the numbers

x, x^logx, y^logy and (xy)^log(xy)

be consecutive terms of a geometric progression?

Title: Re: geometric progression
Post by SMQ on Apr 12^th, 2013, 11:35am

One necessary (but not sufficient) condition: x^{1 + log y} = y^{-log x}

--SMQ

Title: Re: geometric progression
Post by pex on Apr 12^th, 2013, 2:08pm

I find only three solutions: (x,y) = (1,1) or [hide]( exp((1 (+/-) sqrt(3)) / 4) , exp(-1/2) )[/hide], assuming the "log"s are natural logs.
Hint: your numbers are in geometric progression if and only if [hide]their logs are in arithmetic progression[/hide].