This is related to organization set up. i just faced the issue. Please see this . If you have record type in task and event and set up is referring master under user interface -> global action the button wont appear.
New Event, Log a Call, New Task are missing from the Activity related actions on the Activity tab in Lightning
Knowledge Article Number 000247872
When viewing the Activity composer related list, you may notice that some activity actions within the lightning experience do not appear on records even though those actions have been added to the Salesforce for iOS and Android and Lightning Experience Actions section.
-Log a Call
If any of the actions above are missing and you have checked the page layouts to ensure they have been added, then it may be related to not associating a record types with any of those 3 specific global actions.
To confirm this,
1) -In the Lightning Experience, Setup Home | User Interface | Global Actions | Global Actions
-In Aloha Classic, Setup | Global Actions
2) Click Edit next to any of the missing global actions (New Event, New Task, Log a Call)
3) Look at the 'Record Type' pick list and see if the selected value is Not one of your custom record types or if it says --Master--
To resolve the issue, you will have up update the Record Type to be set to a default value that is one of the custom record types. --Master-- will not work.
Note: If you have no custom record types for these objects (Tasks, Events) then you won't be able to select any Record Type. If you do have custom record types, it needs to be associated with the global action to work in the lightning experience under the Activity composer.