How to find $x$ given $\log_{9}\left(\frac{1}{\sqrt3}\right) =x$ without a calculator? I was asked to find $x$ when: $$\log_{9}\left(\frac{1}{\sqrt3}\right) =x$$ 
Step two may resemble: 
$${3}^{2x}=\frac{1}{\sqrt3}$$
I was not allowed a calculator and was told that it was possible. I put it into my calculator and found out that $x$=-0.25 but how do you get that?

I think the key here is to remember what the logarithm actually means: $\log_a(b)$  is the answer to the question "which power of $a$ equals $b$?"  In your case you need to answer the question "which power of nine equals $1/\sqrt{3}$?"  The answer to this is not too difficult to find: $9^{-1} = 1/9$ and taking square roots twice gives the answer.
