We show that if $G$ is a group of exponent 5, and if $G$ satisfies the Engel-4 identity $[x,y,y,y,y]=1$, then $G$ is locally finite. By a result of Traustason, this implies that Engel-4 5-groups are locally finite. We also show that a group of exponent 5 is locally finite if and only if it satsifies the identity
$$[x,[y,z,z,z,z],[y,z,z,z,z]]=1.$$
This result implies that a group of exponent 5 is locally finite if its three generator subgroups are finite.
1991 Mathematics Subject Classification: 20D15, 20F45, 20F50.