Question

How do you evaluate `log_8 (-1/64)`?

Answer 1

As stated, there is no answer since there is no real number to which 8 can be raised to give a negative number. The following equations are true, however, if you are interested: `log_{8}(1/64)=-2` and `log_{8}((1/64)^{-1})=2`.

Explanation:

By definition, for `x>0`, `log_{8}(x)` is the unique real number satisfying the condition that `8^{log_{8}(x)}=x`. Since `8^{y}>0` for all real `y`, the quantity `log(-1/64)` has no real number answer.

For the other examples given, since `8^{-2}=1/(8^2)=1/64` and `8^{2}=64`, it follows that `log_{8}(1/64)=-2` and `log_{8}((1/64)^{-1})=2`.