incorrect calculation of Log[Exp[...]]

Questions
,
1 
Log[Exp[(5*I*\[Pi])/3]] - (I*5 \[Pi])/3]

It get the answer -2?i. However, it must be zero. And it also cause the incorrect answer for Residue

Residue[Log[z]/(1 + z^3), {z, E^((I*5 \[Pi])/3)}]

How do I modify my input?

2

“Log[Exp]” does not always equal “a” even though Log and Exp are supposed to be inverse functions. This is because of complex branch cuts. In fact, “Log[Exp]” can be off from “a” by a constant factor, in this case a multiple of “2 Pi I”.