Kevin,
I have already been down the route you need to take to get your machine working accurately again.
Because it is such a rather old lathe (mine is even older), Stans suggestion of a 3MT ER collet chuck is a good one, except with a lathe this old, the spindle hole just might be a bit off wack and worn, in which case it wouldn't be any better than your chuck is. To try it out, clean out the taper, wack a taper up the spout and stick a clock on it, if it is accurate enough for your needs, take Stans advice, otherwise, see below.
Stans suggestion about about grinding the jaws is a valid point on this score, IMHO just not worth the hassle of doing.
Two ways around your problem of getting an accurate method of workholding.
The first is to get your spindle taper reground or machined while it is in the machine, using the machine itself, (don't use reamers, that will only follow the old taper, been there, got the tee shirt). This will allow you to follow the route suggested by Stan and get an accurate but limited workholding.
The second method is to forget about the spindle taper (fix that later) and buy a few screw on backplates, then machine them to fit almost any chuck you require, 5C collet chuck, new 3 jaw etc.
Doing it this way will ensure that everything is spot on accurate to your spindle as it is fitted in the machine. The only thing to worry about then is how accurate the new chuck is, a fairly cheap 3 jaw can have as much as 3 thou runout from new, so always tighten using the same key hole (usually marked on the side of the chuck), because this is the key hole they used when the original grind was done. If there isnt a marking, tighten up in turn using each key hole and find which one gives the least runout, and then mark that hole for permanent use. I have found that four jaw self centreing chucks a lot more accurate. Also consider soft jaws for your self centreing chucks, you can machine them very accurately and have no runout at all.
I fitted a new, cheap 5c collet chuck to a new backplate and even that had 1/2 thou runout (in the chuck itself), which I corrected myself by using a toolpost grinder in the chuck jaw. No runout at all now. 5c collets are only guaranteed accurate to 1/2 thou anyway.
In the UK you can buy backplates ready made for lathes to do just this sort of thing.
Your lathe uses the same size as mine does.
LMS do the same thing
http://littlemachineshop.com/info/spindles.php
These people might be able to help you as they still support all the old Logan lathes.
http://www.loganact.com/
I hope this has helped, and not sent you on a journey of empty wallet and heartbreak.
Precision usually costs.
John
