Solution

Find $\text{ord}_{19} 8$.


We simply compute positive integer powers of $8\pmod{19}$ and look for the least positive exponent that results in a power that is congruent to $1\pmod{19}$.

$$\begin{array}{rcl} 8^1 &\equiv& 8 \pmod{19}\\ 8^2 &\equiv& 7 \pmod{19}\\ 8^3 &\equiv& 18 \pmod{19}\\ 8^4 &\equiv& 11 \pmod{19}\\ 8^5 &\equiv& 12 \pmod{19}\\ 8^6 &\equiv& 1 \pmod{19} \end{array}$$

Since $6$ is the first such exponent that works, $\text{ord}_{19} 8 = 6$.