Question

Suppose \(A\) and \(B\) are positive real numbers for which \(\log_A B = \log_B A\). If neither \(A\) nor \(B\) is 1 and \(A \neq B\), find the value of \(AB\).

Answer 1

let \[\log_AB = \log_B A = k\]

\(\implies B = A^k ~~\text{and} ~~A = B^k\tag{1}\)

\(\implies AB = A^kB^k=(AB)^k\)

\(\implies AB=1\) or \(k=1\)

however\(k=1\) is impossible because it gives \(A=B\) from \((1)\)