Is it acceptable and safe to run pip install under sudo?

Use a virtual environment:

$ virtualenv myenv
.. some output ..
$ source myenv/bin/activate
(myenv) $ pip install what-i-want

You only use sudo or elevated permissions when you want to install stuff for the global, system-wide Python installation.

It is best to use a virtual environment which isolates packages for you. That way you can play around without polluting the global python install.

As a bonus, virtualenv does not need elevated permissions.

Is it acceptable & safe to run pip install under sudo?

It's not safe and it's being frowned upon – see What are the risks of running 'sudo pip'? To install Python package in your home directory you don't need root privileges. See description of --user option to pip.

Your original problem is that pip cannot write the logs to the folder.

IOError: [Errno 13] Permission denied: '/Users/markwalker/Library/Logs/pip.log'

You need to cd into a folder in which the process invoked can write like /tmp so a cd /tmp and re invoking the command will probably work but is not what you want.

BUT actually for this particular case (you not wanting to use sudo for installing python packages) and no need for global package installs you can use the --user flag like this :

pip install --user <packagename>

and it will work just fine.

I assume you have a one user python python installation and do not want to bother with reading about virtualenv (which is not very userfriendly) or pipenv.

As some people in the comments section have pointed out the next approach is not a very good idea unless you do not know what to do and got stuck:

Another approach for global packages like in your case you want to do something like :

chown -R $USER /Library/Python/2.7/site-packages/

or more generally

chown -R $USER <path to your global pip packages>