Question

If p = q^r, q=r^p and r=p^q, prove that prq=1.​

Answer 1

Rewrite each given equation as

`p=q^r\implies \log_p(p)=\log_p(q^r)\implies 1=r\log_p(q)\implies r=\frac1{\log_p(q)}=(\ln(q))/(\ln(p))`

`q=r^p\implies \log_q(q)=\log_q(r^p)\implies 1=p\log_q(r)\implies p=\frac1{\log_q(r)}=(\ln(r))/(\ln(q))`

`r=p^q\implies \log_r(r)=\log_r(p^q)\implies 1=q\log_r(p)\implies q=\frac1{\log_r(p)}=(\ln(p))/(\ln(r))`

where each of the last equalities follows from the change-of-base identity,

`\log_m(n)=(\log_b(n))/(\log_b(m))`

for any base b > 0 and b ≠ 1. I picked the natural base, e.

Then

`prq=(\ln(r))/(\ln(q))*(\ln(q))/(\ln(p))*(\ln(p))/(\ln(r))=1`

as required.