It sounds like you want to write to your log file when your log file is read externally, or something to that effect. If that is the case, there is a NotifyFilters value, LastAccess. Make sure this is set as one of the flags in your FileSystemWatcher.NotifyFilter property. A change in the last access time will then fire the Changed event on FileSystemWatcher.
Currently, FileSystemWatcher does not allow you to directly differentiate between a read and a change; they both fire the Changed event based on the "change" to LastAccess. So, it would be infeasible to watch for reads to a large number of files. However, you seem to know which file you're watching, so if you had a FileInfo object for that file, and FileSystemWatcher fired its Changed event, you could get a new one and compare LastAccessTime values. If the access time changed, and LastWriteTime didn't, your file is only being read.
Now, in simplest terms, changes you make to the file while it is being read are not going to immediately show up in the other app, nor are you going to be able to "get there first", lock the file and write to it before they see it. So, you cannot use FileSystemWatcher to "intercept" a read request and show the content you want that app to see. The only way the user of another application can see what you just wrote is if the application is also watching the file and re-loads the file. That will fire another Changed event, causing an infinite loop as long as the other application continues to reload the file.
You will also get a Changed event for a read and a write. Opening a file in a text editor (virtually any will do), making some changes, then saving will fire two Changed events if you're looking for changes to Last Access Time. The first one will go off when the file is opened by the editor; at that time, you may not be able to tell that a write will happen, so if you are looking for pure read-only accesses to the file then you're SOL.