I don't think this works mod N. Since N is not a prime, you have no guarantee that for k your base, c belongs to the multiplicative group generated by k.

Take for instance N = 15, you can't find a log base 12 of 7, for instance (12 -> 9 -> 3 -> 6 -> 12). So you'd first need to find a fitting base, not sure how hard this is.

It has been proven [0] that factoring is as hard as solving DLP, but RSA also relies on the modular e-th root hypothesis, which we don't know to be as hard or easier than factoring.

In the end, because of [0] is you solve DLP you solve factoring yes, but we usually don't talk about DLP for RSA, even if as I said I was nitpicking.

[0] https://www2.eecs.berkeley.edu/Pubs/TechRpts/1984/5973.html