Date time cast preferred method
My personal preference is to never use BETWEEN since its not abundantly clear, from the code itself, whether the range includes 2019-01-01 and 2019-12-31.  I realize one can easily check the manual to see how BETWEEN operates, but in my opinion, this is far more easily read, without any chance for confusion:

SELECT LogId
FROM [Master].[dbo].log mlog
WHERE cast(LogDate as Date) &gt;= '2019-01-01' 
    AND cast(LogDate as Date) &lt;= '2019-12-31'


Be aware, though, that cast(LogDate as date) is not an optimal way to perform this query.  The far better solution would be:

SELECT LogId
FROM [Master].[dbo].log mlog
WHERE LogDate &gt;= '2019-01-01T00:00:00.000' 
    AND LogDate &lt; '2020-01-01T00:00:00.000'


The above statement is SARGable, and will also ensure you get the range desired.  Comparing &lt;= '2019-12-31' actually means before December 31st, 2019, when you probably actually mean up until the end-of-day on December 31st, 2019.  Looking for dates less than the start of the next day ensures you get all the rows between the start of the day on 2019-01-01 and the end of the day on 2019-12-31.

Note, I'm including a T between the date and time components; this indicates to SQL Server to treat the value as an ISO8601 date, which will prevent possible errors that might be introduced under non-US English language installations.