It is sensible to write down clearances and as much information as you can. We do not inspect the flight logs but 1st officers are expected to write down all information so we can use it and refer to it.
JAR OPS 1.1055 does require you to write down initial and subsequent flight clearances concurrent and permanaent in nature.
Whether a frequency change is a clearance is a debate to which I am sure there is no end.
MM