Logarithmic topological Hochschild homology of the truncated Brown--Peterson spectrum at height two

We study logarithmic topological Hochschild homology of the truncated Brown–Peterson spectra $\text{BP}n$. We first describe prelog structures on $\text{BP}n$ and analyze the spectral sequences used to compute the $V(n)$-homotopy groups of the associated logarithmic topological Hochschild homology spectra. For general $n$, we establish some structural properties of these spectral sequences that are needed for the computations. We then specialize to the case $n=2$ and carry out the calculation explicitly, determining the $V(2)$-homotopy groups of the corresponding logarithmic topological Hochschild homology spectra.