Is there a way to determine if the following type of equation has an integral solution for both $x$ and $y$?
$$a^x = by +c$$
where $a, b, c$ are constant integers and $a \neq b \neq c$.
I just want to determine if integral solutions exist does not matter if we can know what actually solutions are.
I tried but cannot figure out. I tried to convert the equation to a linear diophantine by taking log of both sides:
$$x\log(a) = \log(by + c)$$
But I do not think that this could help us much.
Exponential Diophantine equations might help but I have not read much about them.
Your question can be restated using modular arithmetic: $$a^x\equiv c\mod{b}$$ This is known as modular exponentiation and is commonly used in cryptography.
As you have correctly guessed, computing possible values for $x$ is quite difficult and modular exponentiation is believed to be a one-way function for most cases.
The values of $c$ often vary in a cyclic fashion. Finding such a cycle will give a list of values of $c$ for which we can find Diophantine solutions, for given $a$ and $b$. However, the one-way nature of the function makes it computationally inefficient to decide such a cycle for large values of $a$ and $b$. So, in my opinion, deciding whether integer solutions exist may be well out of reach for arbitrary values of $a$, $b$ and $c$.
EDIT:
To find such a cycle, the main step is to solve for $x$ in $$a^x\equiv 1 \mod{b}$$ which is computationally difficult as already mentioned. (Further edit: see the Carmichael function)