The Perseus cluster is a nearby cool-core galaxy cluster that hosts an archetypal radio mini-halo. Recent Low Frequency Array (LOFAR) High Band Antenna (HBA) observations at 120-168MHz have revealed the presence of a giant radio halo within the cluster with a size of 1.1Mpc enveloping the mini-halo. By exploring the spectral properties of the radio emission at low frequencies, we can gain deeper insights into the nature of this emission and improve our understanding of its origin. Here we present LOFAR Low Band Antenna (LBA) images of the cluster between 30.0-57.7MHz, with a resolution of 19.2"x15.0" and a r.m.s. noise of 3.7mJy/beam . In our images, we detect both the mini-halo and giant radio halo. We measured the spectral indices between 44 and 144 MHz of the mini-halo and giant radio halo to be -1.34+/-0.10, and -1.01+/-0.11, respectively. An alternative and more direct measurement of the spectrum of the giant radio halo results in a spectral index of -1.28+/-0.15. The discrepancy between both values is caused by the poor ionospheric conditions. In addition, we study two X-ray 'ghost cavities' in the cluster. These cavities are thought to have been produced by an older outburst from the central AGN 3C 84. We measure a spectral index between 44 and 144MHz for the radio plasma in these cavities of -1.86+/-0.12 and -1.90+/-0.12 for the northwest and southern ghost cavities, respectively. Furthermore, by including VLA 352MHz data, we find that the spectrum steepens at higher frequencies. These results are consistent with the ghost cavities being filled with old and aged radio plasma. We also detect the tailed radio galaxies NGC 1265 and IC 310. In our analysis, these sources show signs of spectral steepening along their tails.