How can a log be negative?

How can a log be negative?

A "log" can be negative....we just can't take the "log" of a negative number (or 0)

For instance

Log(.001) = -3

But, Log(-3)....... is undefined......This says, what power can I raise 10 to get "-3"  ???

Answer....there is none !!!!

A log is a power (exponent) and powers can be negative

$$5^{-2}=\frac{1}{25}\\\\ \mbox{therefore}\\\\ log_5 \:\left(\frac{1}{25}\right)=-2$$