Approximation by exponential polynomials

The following exploits the negative integers $k$ as $lambda_k$ (I will go for the simpler, not necessarily for the best bounds, and make it explicit). Take a uniform approximation by polynomials $p_m$ for the function $f(t):=logleft(frac{1}{1-t}right)$ on the interval $J:=[0,1-1/e]$. Assume further $0leq p_m(t)leq 1$ for all $tin J$ (you can always get this). Since
$$epsilon_m:=|f-p_m|_{infty,\,J}=o(1),$$
as $mto infty,$ we get, putting $t:=1-e^{-x}$, that $|x-p_m(1-e^{-x})|_{infty,[0,1]}=epsilon_m$ and, more in general, a bound on the uniform distance on [0,1] between the polynomial $q(x)$ and the exponential polynomial
$$q_m(x):=qleft(\,p_m(1-e^{-x})right).$$
Indeed, by the mean value theorem applied to $q$ and the two points $x$ and $p_m(1-e^{-x}),$ both belonging to the interval $[0,1],$
$$ | q - q_m |leq \|q'|_{infty,\, [0,1]}\, epsilon_m .$$

For instance, the Taylor expansion of order $m$ of $f$ gives
$$p_m(t):=sum_{k=1}^{m}frac{t^{\,k}}{k}$$
with $0leq p_m(t)leq 1$ for all $tin J$ as said, and with $epsilon_mleqfrac{1}{m+1}.$