If you only want to listen to the 160MHz railroad traffic, pretty much any VHF antenna will work. If you are looking for something to replace a log periodic, you could just go with a VHF Yagi beam. The log periodic antenna are broadbanded, which can be useful, but if you are only listening to a 1MHz wide piece of spectrum, you don't need one. A 3 element VHF Yagi would probably work quite well and provide more gain than a simple vertical. But, as OZ said, an omnidirectional would be best if you are looking for coverage from all points.
Building your own Yagi is quite easy and there are a lot of plans on the internet. A new commercial 3 element Yagi is going to probably start in the upper $100 range, and go higher. An amateur grade 2 meter Yagi would work well, and be cheaper.