Merriam-Webster's Collegiate Dictionary:
Main Entry: like
2 : in the same way that : as ‹they raven down scenery like children do sweetmeats— John Keats›
3 a : in the way or manner that ‹the violin sounds like an old masterpiece should› ‹did it like you told me›
[In one of its (infrequent) usage notes]
usage Like has been used as a conjunction since the 14th century. In the 14th, 15th, and 16th centuries it was used in serious literature, but not often; in the 17th and 18th centuries it grew more frequent but less literary. It became markedly more frequent in literary use again in the 19th century. By mid-century it was coming under critical fire, but not from grammarians, oddly enough, who were wrangling over whether it could be called a preposition or not. There is no doubt that, after 600 years of use, conjunctive like is firmly established. It has been used by many prestigious literary figures of the past, though perhaps not in their most elevated works; in modern use it may be found in literature, journalism, and scholarly writing. While the present objection to it is perhaps more heated than rational, someone writing in a formal prose style may well prefer to use as, as if, such as, or an entirely different construction instead.