A couple of comments. In Mathematica
log (x)
means the symbol log multiplied by x. Also log is not the logarithm function, Log is. So if the latter is what you meant, then the proper way to express Log of the argument x is
Log[x]
Second comment" in making use of a parameter like k there is no need to specify that it has the Constant Attribute since it has no value and therefor has no dependence on anything.
Final comment: The function Series is meant to be evaluated with an explicit integer in the place where you have k entered. That is the meaning of the error message that you received.
So you can certainly ask to perform an evaluation like
Series[Log[x], {x, 1, 5}]