Differentiating Riemannian logarithmic map Let $(M,g)$ be a Riemannian manifold, geodesically complete, and assume logarithms are well defined and smooth. 
Let $c: I\to M  $ be a smooth path in $M$, and $x\in M$. Can we say something about $$\Vert\nabla_{\dot{c}(t)}\log_{c(t)}(x)\Vert_{c(t)} ?$$
I can easily prove that $$\Vert\nabla_{\dot{c}(t)}\log_{x}(c(t))\Vert_{x}=\Vert \dot{c}(t)\Vert_{c(t)},$$ by differentiating the equality $$\exp_{x}(\log_x(c(t)))=c(t),$$ and using the Gauss lemma to justify that $d(\exp_x)_{\log(c(t))}:T_x M\to T_{c(t)}M$ is a radial isometry. 
Is it also true that $$\Vert\nabla_{\dot{c}(t)}\log_{c(t)}(x)\Vert_{c(t)}=\Vert \dot{c}(t)\Vert_{c(t)}?$$
 A: I believe the answer to your question is "no".  I am not familiar with the term "Riemannian logarithmic map," but I imagine you mean the inverse of the Riemannian exponential map.  Your condition is thus that the exponential map from any point of $M$ is a diffeomorphism (in particular this implies that $M$ is diffeomorphic to $\mathbb{R}^n$).
Since the exponential map from $x$ is a diffeomorphism, there exist geodesic normal co-ordinates about $x$ for the whole manifold, and so the corresponding distance function $r=d(x,\cdot)$ is smooth on the whole manifold (except at $x$).
At each point $y$ in $M\setminus\{x\}$, one has that $\operatorname{exp}_y(-r\nabla r) =x$.  So, in your terminology, $\operatorname{log}_y(x)=-r\nabla r$.
You ask essentially whether, at each point $c$ in $M$, for each vector $v$ in $T_cM$,
$$\lVert\nabla_{v}(-r\nabla r)\rVert_c=\lVert v\rVert_c;$$
equivalently whether the map from $T_cM$ to $T_cM$,
$$v\mapsto \nabla_v(-r\nabla r),$$
is always an isometry (of vector spaces).  In general it is not.  For example, in the hyperbolic plane, if I calculate correctly, 
$$\nabla_{\frac{\partial}{\partial\theta}}(r\nabla r)=r\coth(r)\frac{\partial}{\partial\theta}.$$
This is closely related to the famous question of the "Hessian of the square of the distance function" (since $\nabla (-rdr)=-\tfrac{1}{2}\operatorname{Hess}(r^2)$), which is equal to the identity only in flat manifolds.
