Matrix functions have become a central topic in linear algebra, and many problems of their numerical approximation appear often in scientific computing. This thesis concerns with matrix functions times a vector with a special attention in the matrix logarithm case. In many applications the original matrix may be large, sparse or structured. In this case evaluating the matrix function times a vector by first computing the full matrix function is usually unfeasible, so that it has sense to approximate the solution saving storage and computational time. Looking into the literature in numerical linear algebra, the standard approach for computing the matrix function times a vectors directly is based on a polynomial Krylov subspace approach that only requires matrix–vector products of the original matrix. This project deals with rational Krylov subspace which have been used recently in this context though it was originally presented for eigenvalue problem in the 90s.