It also interests me, whether your \( C^{\infty} \) slog solution is even analytic.
This could be numerically supported by computing the at 0 developed series \( t_b(x) \) (which was originally meant to be defined on \( (0,1) \)) at some \( x\in(1,2) \) (if convergent there, what is anyway the convergence radius of your slog series?) and then compare the result with \( t_b(\log_b(x))+1 \). Or vice versa to compare \( t_b(x) \) for some \( x\in(-\infty,0) \) with \( t_b(b^x)-1 \). If they are all equal its quite probably analytic.
This could be numerically supported by computing the at 0 developed series \( t_b(x) \) (which was originally meant to be defined on \( (0,1) \)) at some \( x\in(1,2) \) (if convergent there, what is anyway the convergence radius of your slog series?) and then compare the result with \( t_b(\log_b(x))+1 \). Or vice versa to compare \( t_b(x) \) for some \( x\in(-\infty,0) \) with \( t_b(b^x)-1 \). If they are all equal its quite probably analytic.