Logs can absolutely become the bottleneck for data modification operations, yes.

Information written to the log is both more complex and simpler than the T-SQL. It's not a straight copy of the T-SQL. Instead it records the page numbers and data pages of operations, data values added or removed, information about the transaction, start and stop times of the transaction and whether or not the transaction completed. Not to mention the Log Sequence Number. So, writing 'Dog' to a column, more than 'Dog' is stored in the log. But, it is an optimized process, so it's not usually the bottleneck (but certainly can be).

Funny, I'm just updating the locking and blocking chapter in my book, so I'm going through all the ACID properties and that's a big part of the transaction log use in SQL Server, supporting that aspect of relational storage.