There is obviously no closed-form solution. Need to solve numerically after assigning values to a and b, for example
In[10]:= a = 2; b = 3; FindRoot[
Log[Sqrt[(1 + x + b*x^2)]] == (1 + x + a*x^2), {x, 1}]
During evaluation of In[10]:= FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances. >>
Out[10]= {x -> -0.422306}
