If log B base A = log A base B, then what is the value of (A)(B)? A and B are positive real numbers, neither A nor B is 1, and neither are equal.
Use the base change formula:
$\displaystyle \log_b(a)=\dfrac{\ln(a)}{\ln(b)}$
Therefore:
$\displaystyle \log_b(a)=\log_a(b)~\implies~ \dfrac{\ln(a)}{\ln(b)} = \dfrac{\ln(b)}{\ln(a)}~ \implies~(\ln(a))^2=(\ln(b))^2$
$\displaystyle (\ln(a))^2=(\ln(b))^2~\implies~(\ln(a))^2-(\ln(b))^2=0~\implies~$ $\displaystyle \left(\ln(a)-\ln(b)\right) \left(\ln(a)+\ln(b)\right)=0$
$\displaystyle \ln(a)=\ln(b)~\implies~ a= b$ (this case is excluded!) or
$\displaystyle \ln(a)=-\ln(b)~\implies~ a= \dfrac1b$
Therefore:
$\displaystyle \boxed{a \cdot b = 1}$