  Let $\mathscr{L}$ denote the $\mathbf{Q}$-vector space of logarithms of
algebraic numbers. In this expository work, we provide an introduction to the
study of ranks of matrices with coefficients in $\mathscr{L}$. We begin by
considering a slightly different question, namely we present a proof of a weak
form of Baker's Theorem. This states that a collection of elements of
$\mathscr{L}$ that is linearly independent over $\mathbf{Q}$ is in fact linear
independent over $\overline{\mathbf{Q}}$. Next we recall Schanuel's Conjecture
and prove Ax's analogue of it over $\mathbf{C}((t))$.
  We then consider arbitrary matrices with coefficients in $\mathscr{L}$ and
state the Structural Rank Conjecture, which gives a conjecture for the rank of
a general matrix with coefficients in $\mathscr{L}$. We prove the theorem of
Waldschmidt and Masser, which provides a lower bound giving a partial result
toward the Structural Rank Conjecture. We conclude by stating a new conjecture
that we call the Matrix Coefficient Conjecture, which gives a necessary
condition for a square matrix with coefficients in $\mathscr{L}$ to be
singular.
