The $k^{-1}$ spectral law for near-wall turbulence has received only limited experimental support, the most convincing evidence being that of Nickels et al. (Phys. Rev. Lett. vol. 95, 2005, 074501.1). The real-space analogue of this law is a logarithmic dependence on $r$ of the streamwise longitudinal structure function. We show that, unlike the $k^{-1}$ law, the logarithmic law is readily seen in the experimental data. We argue that this difference arises from the finite value of Reynolds number in the experiments. Reducing the Reynolds number is equivalent to restricting the range of eddy sizes which contribute to the $k^{-1}$, or ln$r$, laws. While the logarithmic law is relatively insensitive to a truncation in the range of eddy sizes (it continues to hold over the relevant range of eddy sizes), it turns out that the $k^{-1}$ law is not. This is a direct consequence of the so-called aliasing problem associated with one-dimensional spectra, whereby energy is systematically and artificially displaced to small wavenumbers.