Through the study of elliptic units, the theory of complex multiplication allows one to understand explicit class field theory for quadratic imaginary fields. By replacing complex analysis with its $p$-adic counterpart, we introduce a proposal for a theory of ``real multiplication.'' Using the modular symbols attached to a modular unit $\alpha$ of level $N > 1$,we define certain numbers $u \in \C_p^\times$ which are intuitively analogous to the classical elliptic units arising from $\alpha$. In this vein, we conjecture that the elements $u$ belong to specific abelian extensions of real quadratic fields. Although this conjecture is still open, we are able to analyze the analytic properties of the numbers $u$. We prove a specific formula relating the $\ord_p$ and $\log_p$ of $u$ to certain partial zeta functions (classical and $p$-adic, respectively). The second formula is analogous to the classical Kronecker Limit Formula. The existence of a unit satisfying the properties we demonstrate is the $p$-adic Gross-Stark conjecture; thus our construction gives an analytic construction of Gross's unit, minus a proof of its algebraicity.