Chebyshev economization of a logarithm Taylor polynomial ($p_3=1/32+15x/16-3x^2/4+7x^3/12$)

Deleting the degree-four and degree-five Chebyshev components of $x-x^2/2+x^3/3-x^4/4+x^5/5$ gives the displayed cubic. The deleted terms are $-T_4/32+T_5/80$, whose supremum norm on $[-1,1]$ is exactly $7/160$, attained at $-1$. This bounds the additional polynomial change. It does not make the total uniform error for $\log (1+x)$ finite near its singular endpoint $-1$.