A simple log that tracks hours worked and and a python script that calculates wages
I think regex is a bit much for this scale of project. I decided to try rewriting your function without using regex, and I think the result is much cleaner and more Pythonic.

In addition, I replaced your hours_full  list of lists with a single number that tracks the total. This greatly simplifies the code.

However, the general structure is essentially the same.    

def wage_calculator(log_txt_file, month, year, wage):
    date_pattern = "**Date**: "  # pattern identifying the date in the file
    hours_pattern = "**Hours**: " # pattern identifying the hours in the file
    hours_total = 0.0

    with log_txt_file as f:
        for line in log_txt_file:
            if line.startswith(date_pattern):
                if month in line and str(year) in line:
                    f.next() # skip a line
                    # skip the start of the line to just get a list of hours
                    hours_line = f.next()[len(hours_pattern):]
                    hours = sum(map(float, hours_line.split(', ')))

                    hours_total += hours

                    print line.rstrip()
                    print "Hours logged: " + hours_line
                    print "Total hours for the day " + str(hours) + "\n"

    print "Total hours worked in {0} {1} is {2}".format(month, year,
                                                        hours_total)
    print "At ${0}/hr, your total wage for {1} {2} is ${3}".format(wage, month
                                                                   , year, hours_total * wage)


Beyond this, I would look into more error handling and data cleanup. For example, verify that the **Hours** line matches the **Time** line.

Also a .txt doesn't seem suitable for this data, maybe use a .csv?