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We study analytically the logarithmic corrections to the critical exponentsof the critical behavior of correlation length, susceptibility and specificheat for the temperature and the finite-size scaling behavior, for a generic$\phi^3$ theory at its upper critical dimension (six). We have also computedthe leading correction to scaling as a function of the lattice size. Wedistinguish the obtained formulas to the following special cases: percolation,Lee-Yang (LY) singularities and $m$-component spin glasses. We have comparedour results for the Ising spin glass case with numerical simulations finding avery good agreement. Finally, and using the results obtained for the Lee-Yangsingularities in six dimensions, we have computed the logarithmic correctionsto the singular part of the free energy for lattice animals in eightdimensions.Comment: 18 pages. We have extended the computation to lattice animals in  eight dimensions. To be published in Journal of Physics

Logarithmic corrections for spin glasses, percolation and Lee-Yang singularities in six dimensions