Question

why can't a base of a logarithmic function be a negative

Answer 1

the base can be negative, however, if the base is negative, then the output will be complex

Answer 2

of course, once the log function is extended to complex domain, then solutions to the equation a^x=b will be in the form ln(b)/ln(a)+2*c*pi*i/ln(a), since e^(2*pi*i)=1.

Answer 3

so for example, log base (-2) of 16 equals ln(16)/ln(-2). although this is the principle solution, a real solution to (-2)^x=16 can be found by adding 4 pi i/ln(-2). (ln(16)+4 pi i) / ln(-2) = 4(ln(2)+pi i)/(ln(-1)+ln(2)) = 4*(ln(2)+pi i)/(pi i+ln(2)) = 4.