Look at this
c1 = 1.6*10^(-33);
c2 = 5.23*10^(-27);
Table[{x,
(c2/2) + ((1/c1)*(((x^2)/4) + c1)^(1/2)),
Log[(x/2 - ((((x^2)/4) + c1)^(1/2)))/((((x^2)/4) + c1 - x/2)^(1/2))]},
{x,-10,10}]
which returns
{{-10, 3.125*^33, 0.601986402162968 + Pi*I},
{-9, 2.8125*^33, 0.5928118328288697 + Pi*I},
{-8, 2.5*^33, 0.5815754049028404 + Pi*I},
{-7, 2.1875*^33, 0.5674899664194921 + Pi*I},
{-6, 1.875*^33, 0.549306144334055 + Pi*I},
{-5, 1.5625*^33, 0.5249110622493387 + Pi*I},
{-4, 1.25*^33, 0.49041462650586326 + Pi*I},
{-3, 9.375*^32, 0.43773436867694987 + Pi*I},
{-2, 6.25*^32, 0.3465735902799726 + Pi*I},
{-1, 3.125*^32, 0.1438410362258906 + Pi*I},
{0, 2.5*^16, 0. + Pi*I},
{1, 3.125*^32, Indeterminate},
{2, 6.25*^32, Indeterminate},
{3, 9.375*^32, Indeterminate},
{4, 1.25*^33, Indeterminate},
{5, 1.5625*^33, Indeterminate},
{6, 1.875*^33, Indeterminate},
{7, 2.1875*^33, Indeterminate},
{8, 2.5*^33, Indeterminate},
{9, 2.8125*^33, Indeterminate},
{10, 3.125*^33, Indeterminate}}
along with warnings about a division by zero and 0.*ComplexInfinity