Isometries of generalized $log$-spaces

In this paper studied isometries of $F$-spaces of integrable functions with logarithm. In particular, using passports of Boolean algebra, a necessary and sufficient condition of isometry $F$-spaces of integrable functions of logarithm with respect to strictly positive $\sigma$-finite measures is proved. In this work,external, internal and generalized $log$ algebras are considered separately.