The discrete logarithm

Powers.
- $2^1=2$
- $2^2=4$ ✓
- $2^3=8$
- $2^4=16\equiv5$
- $2^5=10\equiv-1$
- $2^6\equiv-2\equiv9$
- $2^7\equiv-4\equiv7$
- $2^8\equiv-8\equiv3$
- $2^9\equiv6$
- $2^{10}\equiv1$

Solution. $k=2$.

Conclusion. $\log_2 4=2$ in $\mathbb{F}_{11}^*$. Note that $2$ is a primitive root: it generates the whole group of order $10$.