Shell Programming and Scripting

Is it possible to store all standard-out of a bash script and the binaries it calls in a log file AND still display the stdout on screen?

I know this is possible to store ALL stdout/stderr of a script to a single log file like:
exec 1>&${logFile}
exec 2>&1

But running a script with the above would make the script absolutely silent from the commandline, I was hoping for something like:

exec 1>& | tee -a ${logFile}
or
exec 1>& tee -a ${logFile}


But both of those fail.

And no, I don't have the option of modifying every outputable line in the script to have a (| tee -a ${logFile})

A workaround of sorts I'm making use of while debugging:

Code:

sh script &
tail -f log

(with the script writing everything to the log file, of course).

I have seen some solutions with hacks internally around the "tail -f" method, but it gets messy fast. Any other ideas?