Depends on the cloud environment you're using but typically it makes sense to put them on the same logical drive. If you need additional IOPS you can stripe across multiple volumes but still present a single drive.
In a cloud environment you are not the sole consumer of the storage you are allocated. You get a small slice of a very large pie and as such all IO is random in nature. There is nothing to be gained from trying to separate sequential access (log) from random (data).
Nothing to be gained from the recovery angle either as you can't insist that the two volumes will be allocated from different arrays. Also, there tends to be a different class of protection afforded to the availability of storage from the big players. Azure storage for example is triple replicated within the data centre, with an additional copy replicated to a failover data centre by default.