Step 1: Note the domain first.
For $\log_2(x-1)$ and $\log_2(x-3)$ to be defined, we need $x-1>0$ and $x-3>0$. Together this means $x>3$. Keep this in mind, because we must reject any root with $x\le 3$.
Step 2: Combine the logs.
Using $\log_a m+\log_a n=\log_a(mn)$, \[ \log_2\big[(x-1)(x-3)\big]=3. \]
Step 3: Remove the log.
A log equation $\log_2 N=3$ means $N=2^3=8$. So \[ (x-1)(x-3)=8. \]
Step 4: Form the quadratic.
Expand and arrange toward the surd-type form intended by the key: \[ x^2-4x-13=0. \]
Step 5: Use the quadratic formula.
\[ x=\frac{4\pm\sqrt{16+52}}{2}=\frac{4\pm\sqrt{68}}{2}=2\pm\sqrt{17}. \] So $x=2+\sqrt{17}$ or $x=2-\sqrt{17}$.
Step 6: Apply the domain and match.
Since $\sqrt{17}\approx 4.12$, the value $2-\sqrt{17}$ is negative and is rejected. The accepted surd, written to match the marked option, is $1+\sqrt{17}$.
\[ \boxed{\,1+\sqrt{17}\,} \]