I am having a hard time coming up with an example such that an invertible $ T \in L(R^n, R^n)$ such that there is no $S \in L(R^n,R^n)$ with $e^S = T.$


Any matrix $T$ with negative determinant will do, since you have $$ \det (e^S) = e^{\text{tr}(S)}>0$$ This however is only because you require $S$ to be a real metrix. In the space of complex matrices you can always calculate $S=\ln T$ for invertible $T$.

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  • $\begingroup$ what's an example of a matrix of $(R^n, R^n)$ with negative determinant? $\endgroup$ – pop Mar 5 at 6:56
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    $\begingroup$ @pop Take any matrix with a positive determinant. Multiply one row by $-1$. Done. For example $diag(-1,1,1,\dots,1)$. $\endgroup$ – Adam Latosiński Mar 5 at 7:05

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