Is the set of all polynomials in $\log(x)$ dense in $L^2[0,1]$? Is $\{(\log(x))^k\mid k=0,1,2,\ldots\}$ dense in $L^2 [0,1]$? That is, is the set of all polynomials of logarithm functions dense in the set of square integrable functions on $[0,1]$? 
 A: 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.

A: This is not an answer (and possibly total nonsense), but it's too long for a comment. Let $V=\operatorname{Vect}(1,\log,\log^2,\dots)$, and suppose we wanted to show it is dense (which I suspect). We only need to show that its orthogonal is $0$.
We could try working with a Hilbert basis. I don't know how to calculate the integrals of a cosine/sine-function against $\ln^q$. However, for $p,q\in\mathbb{N}$:
$$\langle t^p,\log^q\rangle=\frac{(-1)^q q!}{(p+1)^{q+1}}$$
Because continuous maps are dense in $L^2[0,1]$ and continuous functions can be uniformly approximated using polynomials, polynomials are dense in $L^2$. Suppose $f\perp V$, and $f\neq 0$. There should be real numbers $c_n,~n\in\Bbb N$ with $f=\sum_{n\in \Bbb N} c_n t^n$ in $L^2$ (which strikes me as perfectly unbelievable). If this holds, letting $c_N$ be the first non-zero coefficient, we'd probably get $$0=\langle f,\ln^q\rangle\sim c_N\frac{(-1)^q q!}{(N+1)^{q+1}}$$ which would be contradictory, and so $V$ would be dense.
Any input appreciated!
