  We present a study of the low-frequency radio properties of star forming (SF)
galaxies and active galactic nuclei (AGN) up to redshift $z=2.5$. The new
spectral window probed by the Low Frequency Array (LOFAR) allows us to
reconstruct the radio continuum emission from 150 MHz to 1.4 GHz to an
unprecedented depth for a radio-selected sample of $1542$ galaxies in $\sim 7~
\rm{deg}^2$ of the LOFAR Bo\"otes field. Using the extensive multi-wavelength
dataset available in Bo\"otes and detailed modelling of the FIR to UV spectral
energy distribution (SED), we are able to separate the star-formation (N=758)
and the AGN (N=784) dominated populations. We study the shape of the radio SEDs
and their evolution across cosmic time and find significant differences in the
spectral curvature between the SF galaxy and AGN populations. While the radio
spectra of SF galaxies exhibit a weak but statistically significant flattening,
AGN SEDs show a clear trend to become steeper towards lower frequencies. No
evolution of the spectral curvature as a function of redshift is found for SF
galaxies or AGN. We investigate the redshift evolution of the infrared-radio
correlation (IRC) for SF galaxies and find that the ratio of total infrared to
1.4 GHz radio luminosities decreases with increasing redshift: $ q_{\rm 1.4GHz}
= (2.45 \pm 0.04) \times (1+z)^{-0.15 \pm 0.03} $. Similarly, $q_{\rm 150MHz}$
shows a redshift evolution following $ q_{\rm 150GHz} = (1.72 \pm 0.04) \times
(1+z)^{-0.22 \pm 0.05}$. Calibration of the 150 MHz radio luminosity as a star
formation rate tracer suggests that a single power-law extrapolation from
$q_{\rm 1.4GHz}$ is not an accurate approximation at all redshifts.
