For example, suppose we have two numbers $x, y \in$ N with their corresponding bases $a$ and $b$
$ x_{a} = y_{b} $
How do you determine $a$ and $b$ such that the equation is satisfied?
I tried interpreting the problem as a equation, such that each digit in my numbers x and y are a coefficient. For example, I have the numbers $241$ and $27$, and my idea was that I try to solve the equation $2a^2+4a+1=2b+7$. Unfortunately, I can't seem to get past that.
For small numbers you can often make progress with divisibility. First note that $a^0=b^0=1$, so in your example we need $$2a^2+4a+1=2b+7\\2a^2+4a=2b+6\\a^2+2a=b+3\\b=a^2+2a-3$$ and you can choose any $a$ you want greater than $4$ and compute $b$. This will duly make $b \gt 7$. When the numbers have three digits you will get quadratic equations, which allow you to require that the square root be integral. Similarly here, if we wanted to start from $b$ we would find $$(a+1)^2=b+4\\a=\sqrt{b+4}-1$$ so we must choose $b$ to be four less that a square. We need $a\gt 4$, so $b \ge 32$