It is well known that for a matrix Lie group the Lie exponential map is $e ^z$. This maps a tangent vector $z$ at the identity to a group element.
On the other hand the general Riemannian exponential map centered at point $x$ is given by $\exp _x \triangle$ which maps a tangent vector $\triangle$ at point $x$ (not necessarily identity element) to a group element.
Is there a relationship between these two exponential maps?
For example is below formula correct? If so, are there any conditions involved?
$\exp _x \triangle = xe ^{x^{-1}\triangle}$