  Assuming a logarithmic interaction between constituent particles, compact and
locally accurate wave functions that describe bound states of the two-particle
neutral and three-particle charged complexes in two dimensions are designed.
Prime examples of these complexes are excitons and trions that appear in
monolayers of Transition-Metal DichalCogenides (TMDCs). In the case of
excitons, these wave functions led to 5-6 correct decimal digits in the energy
and the diamagnetic shifts. In addition, it is demonstrated that they can be
used as zero-order approximations to study magnetoexcitons via perturbation
theory in powers of the magnetic field strength. For the trion, making a
comparison with experimental data for concrete TMDCs, we established that the
logarithmic potential leads to binding energies $\lesssim 25\%$ greater than
experimental ones. Finally, the structure of the wave function at small
distances is established for excitons whose carriers interact via the
Rytova-Keldysh potential.
