I need to know how to do this without a calculator.
log base a (3) = 0.5 is the starting equation. You want to find what log base a (1/9) is.
We can agree that one-ninth is simply 3^-2 (three to the power of negative two), for 1/3^2 would be 1/9. So now we have log base a (3^-2).
If you don't remember this from your logarithm rules, the exponent can be placed to the left of the log. So we go from log base a (3^-2) to [-2 * log base a (3)].
You are told that log base a (3) equals 0.5, so what's 0.5 * -2? Hopefully you know that this is -1 without needing a calculator.