Patrick B. answered • 05/23/19

Math and computer tutor/teacher

pr = q^2

p/r = q^2/r^2

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p^q = q^ r

q^r = r^p

p^q = r^p

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q^r = p^q

r * log q = q * log p

log q = (q/r)* log p

log q = (p/q) * log p

q * log q = p * log p

**q^q = p ^ p**

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similarly

p^q = r^p

q * log p = p * log r

log p = p/q * log r

log p = (q/r) * log r

r * log p = q * log r

**p^r = r ^ q**

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finally

q^r = r^p

r * log q = p * log r

**q^q = p ^ p**

**p^r = r ^ q**

r ^q = r^p

so p^r = r^p = p^q = q^r

Then r=q since p^r = p^q

Let x = r=q.

The given equations become:

p^x = x^x = x^p p/x = x/x = 1 ---> p/x = 1 ---> p = x

Therefore, p=q=r=x

Since they are all the same the identity holds