I compete in couple of local math competitions, and one of the questions asked about solving for a base of x. The question was along the lines of 154x+1043x=143x. This was not thee actual question, as I don't have access to the test, but you should be able to figure out what I mean. Thanks for helping me out!
Solving Base x (contest Q was something like "154_x+1043_x=143_x")
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Dr.Peterson
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The basic technique will be to write each number in expanded form; for example, 154x represents 1*x^2 + 5*x + 4. Put them all together and you will have a polynomial equation (for your example, cubic), which in principle (since you know the solution must be a positive integer) can be solved by factoring or by the rational root theorem.
It can also help to recognize restrictions on the base. In your example, x must be at least 6, since the digit 5 would otherwise be illegal. Sometimes just knowing this allows you to find the answer by trial and error as quickly as you could by algebra.
Actually, I can tell at a glance that your example has no solutions. Can you see why?