I am currently weighing up the advantages and disadvantages between DI and SL. However, I have found myself in the following catch 22 which implies that I should just use SL
Service Locator is an anti-pattern, for reasons excellently described at http://blog.ploeh.dk/2010/02/03/ServiceLocatorIsAnAntiPattern.aspx. In terms of logging, you could either treat that as a dependency just like any other, and inject an abstraction via constructor or property injection.
The only difference with log4net, is that it requires the type of the caller that uses the service. Using Ninject (or some other container) How can I find out the type that is requesting the service? describes how you can solve this (it uses Ninject, but is applicable to any IoC container).
Alternatively, you could think of logging as a cross cutting concern, which isn't appropriate to mix with your business logic code, in which case you can use interception which is provided by many IoC containers. http://msdn.microsoft.com/en-us/library/ff647107.aspx describes using interception with Unity.