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Compare them (Posted on 2018-09-08) Difficulty: 3 of 5
Two positive distinct numbers fulfill the equation ba- ab=1

Which is larger:a or b?

For what (x,y) couple are both terms i.e xy and yx equal?

No Solution Yet Submitted by Ady TZIDON    
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Solution proposed solution | Comment 1 of 7
For part 1, involving a and b, it's impossible to tell whether a or b is larger. Take for example 

In 4^1.8871211878830906589... - 1.8871211878830906589...^4 = 1, b>a
In 2.0940128528581167022... ^4 - 4^2.0940128528581167022... = 1 a>b

Part 2:

x^y = y^x


y/x = log(y)/log(x)

y/log(y) = x/log(x)

This can be solved for a value for the two sides is above e, when, i.e., when x/log(x) has two values.

For example, when Wolfram Alpha is asked to solve x/log(x) = 3, using natural logs, the two exact answers it gives are e^(-W(-1/3)) and e^-W{-1}(-1/3) where {} indicates a subscript, where W is the productlog or Lambert W-function. With a -1 subscript I'm not sure what it means. But 1.8571838602078353365 and 4.536403654973527422 are numerical approximations to x/log(x)=3, so either one raised to the other's power will be the same value, 16.5819024004183....

In fact, 2^4 = 4^2, an integral solution near the above real solution.

  Posted by Charlie on 2018-09-08 13:23:42
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