Does large VARCHAR take up more room in the transaction log
In the case of a small insert, there is no difference in size of transaction log generated.

Here's a demo script to prove it (SQL Server 2016):

--setup
USE master
GO
CREATE DATABASE LogTest
GO
ALTER DATABASE LogTest
SET RECOVERY SIMPLE
GO
USE LogTest
GO

CREATE TABLE dbo.v100 (junk VARCHAR(100) )

CREATE TABLE dbo.v8000 (junk VARCHAR(8000) )
GO

--initial inserts have extra log activity from initializing the table
INSERT dbo.v100
VALUES ('z')

INSERT dbo.v8000
VALUES ('z')

-----------------------------------------------------------------
--ok, let's test this
CHECKPOINT --checkpoint clears log in SIMPLE recovery, which helps limit it to what we want to see
GO

BEGIN TRAN
INSERT dbo.v100
VALUES ('a')
ROLLBACK

SELECT Operation, Context, [Log Record Length], [Log Reserve]
FROM sys.fn_dblog(NULL,NULL)
WHERE Operation NOT LIKE '%CKPT'

CHECKPOINT
GO

BEGIN TRAN
INSERT dbo.v8000
VALUES ('a')
ROLLBACK

SELECT Operation, Context, [Log Record Length], [Log Reserve]
FROM sys.fn_dblog(NULL,NULL)
WHERE Operation NOT LIKE '%CKPT'