Long equation for calculating pH of strong acid solution

The following is the long equation that can be used to calculate pH  for low concentration (<0.00001 M)  strong acid solutions:

      

Where C is the concentration of the acid in Molar units (M).

Keep in mind that the above equation is only worth using if we want accurate calculations for the pH for concentrations of the strong acid that are below 0.00001 M. The following table shows the difference in accuracy between the above equation and the simple equation (pH=-log(C))

The above table shows that using the long equation is useful for very small concentrations of the strong acid and it prevents the pH from being above 7 when very small amount (for example: 10 to the minus 8 M) of acid is in solution. This is due to the fact that at very small concentrations of the acid the ionization of water becomes prominent and can not be ignored.