We consider "positive" switched linear system of odes and propose a new method for the approximation of the upper Lyapunov exponent and lower Lyapunov exponent. The method is based on the iterative construction of invariant positive polytopes for a sequence of discretized systems obtained by forcing the switching instants to form a uniform time grid. These polytopes are then used to generate a monotone piecewise-linear joint Lyapunov function on the positive orthant, which gives tight upper and lower bounds for the Lyapunov exponents. As a byproduct we are able to detect whether the considered system is stabilizable or uniformly stable. Several numerical examples are presented to illustrate the method.