Why do you reject negative base solution for Logs? $log_x64=2$ translates to $x^2=64$
This solves to $x=\pm8$
Why do you reject the solution of $x=-8$ ?
Doesn't it successfully check?
$log_{-8}64=2$ means "The exponent for -8 to get 64 is 2"
which is a true statement, no ?

$\log_{-8}x$ would be an inverse function of $(-8)^x$ but this function does not behave well at all. What would be $(-8)^π$ for example?
