Explicit expression for the connection of 2 kinds of exponential map on Lie group

Question

Suppose I have a Lie group $G$ with a left-invariant metric. Then there are 2 kinds of the map are named exponential maps. One is corresponding to one parameter subgroup, the other is by geodesic from the left-invariant metric. I know that we have an ODE that gives the geodesics on the Lie group by $\dot{g}=ad^*_{\dot{g}}\dot{g}$ where $ad^*$ is the coadjoint operator. However, can I have something more global that connects one-parameter subgroups and geodesics together? I hope to have something like a formula links 2 kinds of exponential or a formula that connects the 2 logarithms together.