First off, hydroxide ion is OH-. H3O+ is actually the hydrogen ion concentration. Ummm...without a calculator, this is a tough one. You can estimate, by first finding the pOH:
pOH = 14-pH = 10.10
Then you can extrapolate into how the log was formed:
So if the pOH = 10 then the concentration of OH- in the solution would be 1E-10 M, and if the pOH = 11 then the concentration of OH- in the solution would be 1E-11.
However, the only way to do this without a calculator is to know the math behind logs and anti-logs...or memorize the tables for values. For example:
Lets say pH = 3.50, you know that if the pH was 3 then the concentration of H+ would be 1E-3 M. Also, if it was 4 then H+ concentration would be 1E-4 M. Here's where memorizing is helpful, once you know the range you can just memorize some numbers:
pH = 3.50; H+ conc = 3.16E-4
pH = 4.50; H+ conc = 3.16E-5
pH = 10.50; H+ conc = 3.16E11
As you might notice, it doesn't matter what the pH is, if it is xx.50 then you can guarantee that the answer will look like 3.16E-xx.
So having a table of these values memorized is helpful and thus a pOH = 10.10 gives you a concentration of about 7.94E-11 M
If you have any more questions or are confused about a certain part then just ask.
Let me clarify something:
As you might notice, it doesn't matter what the pH is, if it is x.50 (where x= 1,2,3,4,5,6,7,8,9,etc) then you can guarantee that the answer will look like: 3.16E-(x+1).