Is the set of all polynomials in $\log(x)$ dense in $L^2[0,1]$?

Let's change the variable: $x=e^{-t}$, $t> 0$. This induces an isometry between $L^2([0,1])$ and the weighted Lebesgue space $L^2((0,\infty),e^{-t})$. The polynomials in $\log x$ become polynomials in $t$. We may decide to orthonormalize them with respect to the weight $e^{-t}$, thus obtaining Laguerre polynomials. The question can now be restated as: do the Laguerre polynomials form a basis of $L^2((0,\infty),e^{-t})$? Interestingly, I could not find an answer to this obvious question in the Wikipedia article, or Mathworld article, or in Springer EOM. At last a Google search brought up this paper where this statement is made explicitly:

It is known that $\{L_n\}$ is a basis in $L^2(0,\infty)$ with respect to the measure $e^{-x}\,dx$ (see, e.g., [6, p. 349]).

[6] G. Sansone, Orthogonal functions, rev. English ed., Dover Publications Inc., New York, 1991.