Today I found myself pondering two series of numbers: One series is the result of $2^x$ (2, 4, 8, 16, etc.) and the other being the result of $3^x$ (3, 9, 27, 81 etc.).
What I initially wondered was: Do these two sets have any common values in them? After thinking about it logically for a moment I concluded that no, these sets must not contain any elements in common simply because the values of the first set are all even and the values of the second set are all odd. It is impossible for there to be a common value between them.
While I was satisfied that I had achieved an answer to my original question, I wasn't quite satisfied with how I got there. I was very lucky that my specific question had such a trivial answer that you can see by just starting to list out the terms, but what would I have done if the answer were not so clear cut? Is there a more systematic way I could have arrived at this answer?
Now, I haven't been in a math class since high school, well over a decade ago, but this felt like something that seemed doable based on what I could remember. I decided to rewrite the question in a way I felt I could reason with:
Consider a number $y$ that can be described by the following two ways: $$y=2^a$$ $$y=3^b$$ Is there a whole number value for $y$ that results in both $a$ and $b$ also being whole numbers?
After working with the problem for a little bit, I suddenly realized that rewriting it like this really didn't help me at all. Even though I knew the ultimate answer to the question was "no", I couldn't seem to determine that more systemically. I could graph these two equations and determine where they intersect (an exersize that helped me realize that there technically is an answer: $y=1, a=0, b=0$. Not a fun answer, but an answer nonetheless.), I could determine that neither function has an asymptote, and thus every single value of $y$ also gives a valid value of $a$ and $b$: What I didn't know how to do was to "filter" those results to only the results that give whole numbers. I don't even know if such a thing is possible.
Even though I know the answer to my original question, I wonder what the "correct" way to solve it would have been had the answer not been so trivial? I imagine that my original instinct to try to make it look like an algebra problem was a mistake, and that the better way to think about it is to think of the numbers as sets, and to describe the sets in such a way that the fact that they contain no common elements would be self-evident. However, I really don't know much about sets and set theory, so I don't know how something like that would normally work, nor how it would look.
So, what would the more general approach to a problem like this look like?