Would the ability to efficiently find Discrete Logs have any impact on the security of RSA? This answer makes the claim that the Discrete Log problem and RSA are independent from a security perspective. 

RSA labs makes a similar statement:


  The discrete logarithm problem bears the same relation to these systems as factoring does to the RSA system: the security of these systems rests on the assumption that discrete logarithms are difficult to compute.


Would the ability to efficiently find Discrete Logs have any impact on the security of RSA?

If you had a discrete log oracle could you use it to attack RSA?
There is a reduction from DL to RSA if the DL oracle accepts composite modulus. For prime modulus, a reduction is not known. I copied the following from this wikipedia page with minor edits. 

Let $n = p\,q$ be an RSA modulus. 


Generate random integer $a$ co-prime to $n$ and a random integer $x$ taken in an interval much larger than $n$, say $[1,1000n]$.
Compute $b = a^x \bmod n$ but don't tell $x$ to the 'discrete log oracle'. 
Instead ask it to find the discrete log of $b \bmod n$ (to base $a$). Let the value returned by the oracle be $y$. 
If $y$ is a solution, then so is $y\pm\varphi(n)$ where $\varphi$ is Euler's totient function, and $0&lt;\varphi(n)&lt;n$. It follows there are at least $1000$ solutions $y$ that lie in the interval of $1000\,n$ integers from which we picked $x$ uniformly at random, thus the oracle has probability $\le\frac1{1000}$ of finding $x$. There is thus a high probability that $y\neq x$. If so, $|x-y|$ will be a multiple of the order of $a$, which can easily be used to factor $n$. I will not go to the details of factoring once (a multiple of) the order of $a$ is known (since this is a well known method, also used in Shor's algorithm). If we are unlucky and $x=y$, then we start over.