(I also added the defenition of logaritm on Elliptic curves)
I should somehow be able to use the information from problem 3b to solve 3c efficiently but I don't know how. Could someone tell me how i could use the information in b to solve c efficient?
Multiplying the second equation by 2 we have the system \begin{align*} 6P+2Q+R=\infty, \\ 4P+2Q+4R=\infty. \end{align*} Subtracting the second from the first gives $2P-3R=\infty$, i.e. $2P=3R$. Since $N=109$, $R = aP$ where $a\equiv3^{-1}\cdot 2\bmod{109}$. As Moo points out, this gives $a\equiv37\bmod{109}$.