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Topic: x^y = y^x What number meaning?
Replies: 12   Last Post: Aug 14, 2017 11:18 PM

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Wally W.

Posts: 269
Registered: 6/15/11
Re: x^y = y^x What number meaning?
Posted: Aug 14, 2017 11:18 PM
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On Sun, 13 Aug 2017 21:32:35 -0500, Serg io wrote:

>On 8/13/2017 6:04 PM, Jim Burns wrote:
>> On 8/13/2017 4:43 PM, Peter Percival wrote:
>>> Mr Sawat Layuheem wrote:
>>
>>>> x^y = y^x What number meaning?
>>>
>>> The solution is
>>> y = -x W(-log(x)/x)/log(x)
>>> where W is Lambert's W function. See
>>> https://en.wikipedia.org/wiki/Lambert_W_function.

>>
>> Also,
>> x^y = y^x
>>
>> x^(y/x) = y
>>
>> x^(y/x - 1) = y/x
>>
>> Let y = m*x
>>
>> x^(m-1) = m
>>
>> x = m^(1/(m-1))
>>
>> y = m*m^(1/(m-1)) = m^(m/(m-1))
>>
>> That gives us lots of examples (x,y), one for each m.
>>
>> ----
>> Hmmm. A slightly prettier answer:
>>
>> k = 1/(m-1)
>>
>> m = (k+1)/k
>>
>> x = ((k+1)/k)^k
>>
>> y = ((k+1)/k)^(k+1)
>>
>> k x y
>> 1 2 4
>> 2 9/4 27/8
>> 3 64/27 256/81
>> ...
>>

>
>that is a nice solution


Indeed.

Keeping some parts separated for speed in a computer program (only one
exponentiation needed):

m = (k + 1) / k

x = m^k

y = x * k

But it loses the beauty of the fractions in Jim's post.




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