when the hydronium ion concentration of a solution is 1x10-10 mole per
liter, what is the pH of the solution? Is the solution acidic or
basic?

The original definition of pH is the negative of the logarithm (base
10) of the hydrogen ion (or hydronium ion) in moles per liter.  Do the
calculation.  If you know what a logarithm is, then you don't even
need a calculator to do this problem.

pH values above 7 are considered basic, while those below 7 are considered acidic.

The pH is estimated from the total hydronium ion concentration. The
total H3O+ concentration have to be calculated. H3O+ comes from two
components, (1) from added acid in water and (2) self dissociation of
water. In this case, total [H3O+] = (10-10) + (10-7) = 1.001x10-7.
Thus the pH will be very close to the nuetral pH, but unmeasureably
low acidity.