The numerical tensor calculus is an efficient tool for treating high-dimensional objects. The tensor formats involve certain representation ranks. The crucial question is the relation between these ranks and the approximation error. The application problem is a diffusion problem whose conductivity coefficient is a log-normal random field. Under suitable assumptions we prove that the approximation error depends only on the smoothness of the covariance function and does neither depend on the number of random variables nor on the degree of the multivariate Hermite polynomials.