A paper (note: SpringerLink, may require subscription) called "Separating Decision Diffie–Hellman from Computational Diffie–Hellman in Cryptographic Groups∗" published in '03 in the Journal of Cryptology has the following quote,
"In 1994 Maurer used a variation of the elliptic curve factoring method to give strong
evidence that CDH and DL are probably equivalent (see ). This approach was formalized
by Maurer and Wolf in  and finally appeared as a journal version in ." (where CDH = "Computational Diffie-Hellman" and DL = "Discrete Log")
Where the relevant citations are included below.
The paper goes on to say "Our goal in this paper is to merge the two approaches
and to construct a family of groups where CDH and DL are equivalent and presumably
hard". Assuming "equivalent" here means CD reduces to DL and DL reduces to CDH, this seem relevant.
So while this isn't a complete answer, the answer might be that at least occasionally we can use a CDH solution to solve a DLog. (Note: I haven't read this paper (yet) so I don't know that they actually met their goal, so this too might be wrong! If someone familiar can confirm whether or not they did, that would be great--I don't have the time or battery life right now to do so.)
I suspect  might shed more light on the issue, if you can find it (I haven't looked), and there may be more recent results.
" U. Maurer. Towards the equivalence of breaking the Diffie–Hellman protocol and computing discrete
logarithms. In Advances in Cryptology -CRYPTO’94, volume 839 of Lecture Notes in Computer Science,
pages 271–281. Springer-Verlag, Berlin, 1994.
 U. Maurer and S. Wolf. Diffie–Hellman oracles. In N. Koblitz, editor, Advances in Cryptology - Crypto
’96, Volume 1109 of Lecture Notes in Computer Science, pages 268–282. Springer-Verlag, Berlin, 1996.
 U. Maurer and S. Wolf. The relationship between breaking the Diffie–Hellman protocol and computing
discrete logarithms. SIAM Journal on Computing, 28(5):1689–1721, 1999."