there are no natural logarithms for negative numbers.
the only way it will work is if you use an imaginary number.
Example:
$${{log}}_{{\sqrt{-{\mathtt{1}}}}}{\left(-{\mathtt{1}}\right)}$$ which is 2, since $${{\sqrt{-{\mathtt{1}}}}}^{\,{\mathtt{2}}} = -{\mathtt{1}}$$