Is there a power of 3 and power of 7 that are equal? I have been trying to find a set of values $(a,b)\in\mathbb Z^+$ which satisfy the following relationship:

$3^a = 7^b$

I haven not been able to find a value that can satisfy this and I also can't find a way disprove it.

Is there a theorem that I am missing?
The theorem you are missing is the fundamental theorem of arithmetic which states that every integer factors uniquely as a product of prime integers.
You can prove that differently as well.

Since $a,b$ are integers, $\frac{a}{b}$ is rational by definition.

But:

$$3^a=7^b \to a\log 3 = b\log 7 \to \frac{a}{b} = \frac{\log7}{\log3}$$ Which is an irrational number. Hence, there are not integers $a,b$ such that: $3^a=7^b$