SQL log file growing exponentially
To answer your question, in FULL recovery mode, growth of the log file is absolutely normal, and of course it will grow to either the limited size or the size of the disk unless it is maintained. The rate of growth will be determined by the type of / frequency of / load of transactions hitting your database. The rate of growth, will also determined by your auto-growth settings, i.e. grow by fixed value or percentage etc.

Logs are typically maintained by performing a log backup at suitable intervals. This truncates the data within the log file itself allowing the space to be re-used for new transactions. 

What is important to note is that the problems you might be facing due to a large log file (i.e. where the size has gotten out of hand due to lack of suitable maintenance) is not necessarily remedied by a transaction log backup alone. You may need to shrink the file to free up disk space, for example.