As I just found out, the idea of Andrew was already considered

by P. L. Walker and others around 1990. I dont know yet exactly were it appeared first but he mentions it in [1]. Even Galidakis' method was mentioned there. (Though everything for base e.)

However till today there seems to be no proof of the convergence (radius) of the slog power series.

[1] P. L. Walker, Infinitely differentiable generalized logarithmic and exponential functions, Mathematics of computation, 57 (1991), No 196, 723-733.

[2] C. W. Clenshaw, D. W. Lozier, F. W. J. Olver and P. R. Turner, Generalized exponential and logarithmic functions, Comput. Math. Appl. 12B (5/6) (1986), 1091-1101.

[3] P. L. Walker, On the solutions of an Abelian functional equation, J. Math. Anal. Appl. 155 (1991), 93-110.

by P. L. Walker and others around 1990. I dont know yet exactly were it appeared first but he mentions it in [1]. Even Galidakis' method was mentioned there. (Though everything for base e.)

However till today there seems to be no proof of the convergence (radius) of the slog power series.

[1] P. L. Walker, Infinitely differentiable generalized logarithmic and exponential functions, Mathematics of computation, 57 (1991), No 196, 723-733.

[2] C. W. Clenshaw, D. W. Lozier, F. W. J. Olver and P. R. Turner, Generalized exponential and logarithmic functions, Comput. Math. Appl. 12B (5/6) (1986), 1091-1101.

[3] P. L. Walker, On the solutions of an Abelian functional equation, J. Math. Anal. Appl. 155 (1991), 93-110.