Find all integer solutions of the equation

Find all integer solutions to $3^a+7=2\cdot5^b$.

Basically I've tried almost every tool I know of NT (Zsigmondy,LTE,reducing to various modulus) but nothing worked.

So far I just know that (1,1) works.

$(1,1)$ and $(5,3)$ are the only solutions.

Suppose we have a solution which is larger than this. Then the RHS is divisible by $5^4 = 625$, and so the LHS must also be.

The order of $3$ modulo $625$ is $500$, and $3^{105} + 7$ is divisible by $625$. So we need: $$a \equiv 105 \pmod{500}$$

Now, the order of $3$ modulo $3001$ is $500$, and $\text{LHS} \equiv 3^{105} + 7 \equiv 1735 \pmod{3001}$. So for such a solution to exist, we need:

$$2\cdot 5^b \equiv 1735 \pmod{3001}$$

But the order of $5$ modulo $3001$ is $250$, and an exhaustive search of the $250$ powers of $5$ modulo $3001$ shows that this cannot happen.

So $(1,1)$ and $(5,3)$ are indeed the only solutions.