Rewrite each given equation as
[tex]p=q^r\implies \log_p(p)=\log_p(q^r)\implies 1=r\log_p(q)\implies r=\dfrac1{\log_p(q)}=\dfrac{\ln(q)}{\ln(p)}[/tex]
[tex]q=r^p\implies \log_q(q)=\log_q(r^p)\implies 1=p\log_q(r)\implies p=\dfrac1{\log_q(r)}=\dfrac{\ln(r)}{\ln(q)}[/tex]
[tex]r=p^q\implies \log_r(r)=\log_r(p^q)\implies 1=q\log_r(p)\implies q=\dfrac1{\log_r(p)}=\dfrac{\ln(p)}{\ln(r)}[/tex]
where each of the last equalities follows from the change-of-base identity,
[tex]\log_m(n)=\dfrac{\log_b(n)}{\log_b(m)}[/tex]
for any base b > 0 and b ≠ 1. I picked the natural base, e.
Then
[tex]prq=\dfrac{\ln(r)}{\ln(q)}\times\dfrac{\ln(q)}{\ln(p)}\times\dfrac{\ln(p)}{\ln(r)}=1[/tex]
as required.
