Solving an exponential equation for a finite field

Question

For a primitive element of the finite field $\alpha$ and $\beta=\alpha^2$, I'm searching for all integer values of $(k,l)$ such that $\beta^l=1+\beta^k$. I have a list of $k$ values which I know are elements of the multiplicative group $\langle \beta \rangle = \langle \alpha^2 \rangle$, but now I must solve for values of $l$ such that $\beta^l=1+\beta^k$ for that given $k$ value. If this equation were over the reals or complex numbers, I could simply take a natural logarithm of both sides and simplify down to get a value for $l$. However, this approach shouldn't work for a finite field. Is there any way to easily find the $l$ value for any given $k$ value without using unnecessarily complicated coding?