The problem of exponential law of steady, incompressible fluid flow in boundary layer and heat transfer are studied in an electrically conducting fluid over a semi-infinite vertical plate assuming the variable thermal conductivity in the presence of a uniform magnetic field. The governing system of equations including the continuity equation, momentum equation and energy equation have been transformed into nonlinear coupled ordinary differential equations using appropriate similarity variables. All the numerical and graphical solutions are obtained through the use of Maple software. The solutions are found to be dependent on three dimensionless parameters including the magnetic field parameter M, thermal conductivity parameter β and Prandtl number Pr. Representative velocity and temperature profiles are presented at various values of the governing parameters. The skin-friction coefficient and the rate of heat transfer are also calculated for different values of the parameters.