Very much I ask to add function of the description of granting of a privilege or the permission. For example, I inserted a command in the / lp menu, user% player% parent set premium, the player buys it for play money, but I can not find out why he got it. Therefore, please add comments (log) to the commands - / lp user% player% parent set premium log: "Purchase privilege for virtual money"

No, this is beyond the scope of the logging functionality in LuckPerms. The action parameter of log entries is limited in terms of characters, and is intended to describe the action (hence the name) and not the cause.

If you want additional logging, then I suggest using an extra plugin which will let you record log entries, and then just post to that log whenever you execute your promotion commands.