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Let $\mathscr{L}$ denote the $\mathbf{Q}$-vector space of logarithms ofalgebraic numbers. In this expository work, we provide an introduction to thestudy of ranks of matrices with coefficients in $\mathscr{L}$. We begin byconsidering a slightly different question, namely we present a proof of a weakform of Baker's Theorem. This states that a collection of elements of$\mathscr{L}$ that is linearly independent over $\mathbf{Q}$ is in fact linearindependent over $\overline{\mathbf{Q}}$. Next we recall Schanuel's Conjectureand prove Ax's analogue of it over $\mathbf{C}((t))$.  We then consider arbitrary matrices with coefficients in $\mathscr{L}$ andstate the Structural Rank Conjecture, which gives a conjecture for the rank ofa general matrix with coefficients in $\mathscr{L}$. We prove the theorem ofWaldschmidt and Masser, which provides a lower bound giving a partial resulttoward the Structural Rank Conjecture. We conclude by stating a new conjecturethat we call the Matrix Coefficient Conjecture, which gives a necessarycondition for a square matrix with coefficients in $\mathscr{L}$ to besingular.

Ranks of matrices of logarithms of algebraic numbers I: the theorems of Baker and Waldschmidt-Masser