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I am trying to prove that $e^x$ is a solution to $f'(x)=f(x)$, and there is a point at which I am concerned that there might be circular logic.

Here's what I've got so far:
$$f'(x)=f(x)$$
$$fracf'(x)f(x)=1$$
$$intfracf'(x)f(x)dx=x+c_0$$
Letting $y=f(x)$ gives
$$intfracdyy=x+c_0$$
Which gives $$ln|y|=x+c_0$$
$$f(x)=c_1e^x$$
QED

The bit I'm concerned about is $intfracdyy=ln|y|$.

Is there any proof of $$intfracdxx=ln|x|$$
which doesn't use $fracddxe^x=e^x$?

Edit: If it wasn't clear, I am asking if there is a way that one can prove that $$intfracdxx=ln|x|$$
without relying on the fact that $fracddxe^x=e^x$?

Here's a way to remove the circularity.

Define $g(x) = int_1^x dt/t$. Then
$$
g(xy) = int_1^xyfracdtt = int_1^xfracdtt + int_x^xyfracdtt = int_1^xfracdtt + int_1^yfracduu = g(x)+g(y)
$$
where $t = xu$ is used for the substitution. This property is unique to logarithm functions, and its base will be the number $e$ such that $g(e) = 1$. So $g(x) = log_e(x)$, and $g^-1(x) = e^x$.

This is one way to define the logarithmic functions.

$$ln x= int _1 ^x fracdtt, text for x > 0 $$

So with this convenience, your proof is OK.

Assuming the question you are asking is indeed that which you are trying to solve, to show that $e^x$ is a solution to $f' = f$ all you have to do is show that $fracddxe^x = e^x$. Which can be fairly easily done by looking at the series definition of $e^x$ term by term and differentiating.

If you're interested in investigating whether or not $ke^x$ is the only solution to $f' = f$ (which it is), then other steps are necessary.

The usual proof of this is by setting $g(x)=f(x)e^-x$ and prove $g'=0$.

See for instance: Proof that $Cexp(x)$ is the only set of functions for which $f(x) = f'(x)$

Nevertheless your reasoning is correct, but solving the ODE this way lacks of rigour (dividing by $f(x)$ without discussing the annulation). Instead it should be used as a hint to find that solutions looks like $Ce^x$ then by CL theorem you claim their maximality and uniqueness.

See for instance LinAlgMan's answer here: https://math.stackexchange.com/a/409974/399263