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We analytically solve the next-to-leading logarithmic Balitsky-Kovchegov equation approximated in the saturation regime, which includes corrections from quark and gluon loops, and large double transverse logarithms. The analytic result for the S -matrix in the saturation regime shows that the exp (- O (Y)) rapidity dependence of the running-coupling solution is replaced by exp (- O (Y 3 / 2)) in the next-to-leading logarithmic solution. The collinearly improved Balitsky-Kovchegov equations are also analytically solved in the saturation regime. This shows that the double collinear logarithms do not contribute to the S -matrix, and the solution is the same as the one obtained from the leading-order Balitsky-Kovchegov equation. The numerical solutions to the leading-order and next-to-leading logarithmic Balitsky-Kovchegov equations are performed in order to test the analytic results derived in the saturation regime. [ABSTRACT FROM AUTHOR]