A log typically describes a part of a tree in its rudiment form, whereas a beam has been crafted into a specific shape for a specific purpose. For example a beam would be used to support a roof or floor in a house. A log would be used in the fire place.

While I was growing up in the US my parents' business was building log homes. A beam is a type of weight-bearing structural unit. Typically a beam spans a horizontal gap and is placed so that its two ends are supported by a vertical structural element such as a column or wall. Horizontal structural elements that are supported on one or both ends by a different horizontal element are called joists. Beams and joists may be fashioned from several different materials including wood, metal or stone. A log is a cut length of the trunk of a tree, either with the bark intact or with the bark removed. A whole log may be shaped to produce a single timber; in some contexts the finished timber may also be called a log. In the context of log home construction, finished timbers that will be placed in layers to form a wall are usually identified as logs. Other finished timbers may be used to span horizontal gaps or other non-vertical gaps and support walls, floors or roofs. Usually these other finished timbers are identified according the their intended structural role, such as beam, joist or rafter.

You call something a log because of what it is. A log is something you cut from a tree. If you cut down a large branch and then clean it up a little bit so that it is a nice straight piece of wood, then we call that a log. But it can't be too small. The smallest thing you would call a log is about the size of your forearm. So a log is a roughly cut piece of a tree which is not too small. If you cut a log into a nice square piece of wood, then it is no longer a log. At that point we call it "lumber". We usually build houses out of lumber. You can also build a house out of logs, but then it would be a very rustic house, like a cabin by a lake. "Log homes" can be really beautiful. You call something a beam because of what it does. A beam supports weight. It is a horizontal member in a building or a bridge (not a vertical member, like a column, or a post). A beam goes over an empty space. It has something to hold it up at either end but not in the middle. You can use a "log" to make a beam, or you can use "lumber" to make a beam. You can also make a beam out of steel, or concrete, or stone, or plastic, or anything that is strong enough to hold up a weight.

Adding to the answers and comments below: The complete form is "log of wood". Put another log on the fire. Beams are structural elements that may be made of an substance that you wish. Logs are wood. "Beam" and "log" also have other meanings, of course that do not overlap.

A log is a tree that has fallen. The bark is still on the outside. A beam is cleaned, smooth, and ready for use. In a room the beams in the ceiling may be visible.