Optical antennas offer unique possibilities for light manipulation on a sub-wavelength scale. Here, we study log-periodic antennas that exhibit broadband directivity as a result of the self-similar relation between the lengths, separations and widths of the elements. We show through numerical simulations that the log-periodic designs have a considerable potential for improvement of both directivity and operation bandwidth over classical Yagi-Uda designs. Moreover, the directivity is more robust against changes in the location of the source or detector at different antenna elements. We systematically study the influence of geometrical parameters on angular performance and local field enhancement to arrive at optimum values. Next, we demonstrate that introducing a gap in the dipole array architecture can provide at least a ten-fold enhancement of the emitted power. Finally we present an optical zigzag antenna capable of both broader spectral response and even higher directivity.