I did not know that, I am now confused about why it is called the discrete logarithm and not discrete division, division and the logarithm are vastly different; since elliptic curves are abelian, the "logarithm" and the "root" are the same problem, this makes no sense, it could as easily be called the discrete root problem.
I think it's called discrete logarithm because it's analogous to the discrete log in a group. I guess technically it should be called discrete division
Elliptic curves are interesting because we can define addition of the curve points and therefore get groups defined over elliptic curves. A point can be added to itself also. That is what leads to generating public keys from private keys in analogy to finite groups. In finite groups we would pick a random number (with some constraints) and then use it as the exponent of the generator of the finite (and cyclic) group. In elliptic curve crypto we also pick a random number (with some constraints) and the multiply (repeatedly add) the generator to itself that many times. The random number is the private key, the result of exponentiation/addition is the public key.
