Take any norm, $\|\cdot\|$on $\mathbb{R}^n,$ and consider the resulting norm on $SL_n(\mathbb{R})$:
$$\|A\|:= sup\{\|Av\|: \|v\|=1\}.$$
Now take any left-invariant Riemannian metric, $g$, on $SL_n$. How do the geodesic balls, $B_g(I, r)$ around the identity matrix, $I$, compare with the metric balls, $B_{\|\cdot\|}(I,r)$ coming from $\|\cdot\|$? In particular do there exist $c, C$ such that $$B_{\|\cdot\|}(I,cr)\subset B_g(I, r) \subset B_{\|\cdot\|}(I,Cr)$$ for all sufficiently small $r$? Or anything of the sort?
I can prove the following. If $h$ denotes the metric on $SL(n,\mathbb{R})$ coming from the operator norm, and $h_1$ denotes a left-invariant metric on $SL(n,\mathbb{R})$, then, at $g \in SL(n,\mathbb{R})$, and for any vector $x$ tangent to $SL(n,\mathbb{R})$ at $g$, we have:
$h(x,x) \leq \|g\|^2 h_1(x,x)$
I just used left translations, and things like that. If interested, I can write some more details. Also, one can prove (using the equivalence of any 2 norms on a finite-dimensional vector space) that there is a $C>0$ such that:
$h_1(x,x) <= C \|g^{-1}\|^2 h(x,x)$.
Hence, if $\|g\|$ and $\|g^{-1}\|$ are bounded above by some constants, then the two metrics induce uniformly equivalent norms on the tangent spaces of that region. In particular, there exists a neighborhood of the identity in $SL(n,\mathbb{R})$ for which the two geodesic distances are equivalent.