The maximum-term method is a consequence of the large numbers encountered in statistical mechanics. It states that under appropriate conditions the logarithm of a summation is essentially equal to the logarithm of the maximum term in the summation.
These conditions are (see also proof below) that (1) the number of terms in the sum is large and (2) the terms themselves scale exponentially with this number. A typical application is the calculation of a thermodynamic potential from a partition function. These functions often contain terms with factorials  which scale as  (Stirling's approximation).