1. Given the equation $x \log_2 x + \log_2 (x - 3) = 2$, we need to find the value of $x$. 2. Recall that $\log_2 (x - 3)$ is defined only if $x - 3 > 0$, so $x > 3$. 3. Rewrite the equation for clarity: $$ x \log_2 x + \log_2 (x - 3) = 2 $$ 4. Use the logarithm property to combine terms if possible, but here we cannot combine directly because of the multiplication by $x$. 5. To solve, test possible values based on domain $x > 3$. 6. Let’s use substitution: let $y = \log_2 x$, then $x = 2^y$. 7. Substitute $x$ in $x - 3$ as $2^y - 3$. 8. The equation becomes: $$ 2^y \cdot y + \log_2 (2^y - 3) = 2 $$ 9. Now check integer values of $y$ such that $2^y - 3 > 0$, i.e., $2^y > 3$. 10. Try $y=2$: then $2^2=4$, $4 \cdot 2=8$, $\log_2 (4 - 3) = \log_2 1=0$, so sum is $8 + 0=8 \neq 2$. 11. Try $y=1.5$: $2^{1.5} \approx 2.828$, so $2.828 \cdot 1.5 = 4.242$, $\log_2 (2.828 - 3)$ is not defined (negative number). 12. Try $y=1$: $2^1=2$, $2 \cdot 1=2$, $\log_2 (2 - 3) = \log_2 (-1)$ undefined. 13. Try $y=3$: $2^3=8$, $8 \cdot 3=24$, $\log_2 (8 - 3) = \log_2 5 \approx 2.32$, sum is $24 + 2.32=26.32$. 14. Since testing values manually is tedious, let's check $y$ near $1.8$: At $y=1.8$, $2^{1.8} \approx 3.482$, $3.482 \times 1.8 = 6.268$, $\log_2 (3.482 - 3) = \log_2 0.482$ is negative, so sum less than 2. 15. Let's find solution by rewriting: Use properties of logs to rewrite original equation. 16. Note $\log_2 (x - 3) = 2 - x \log_2 x$ from original equation. 17. Exponentiate both sides with base 2: $$ x - 3 = 2^{2 - x \log_2 x} $$ 18. Rewrite $2^{ - x \log_2 x } = (2^{\log_2 x})^{-x} = x^{-x}$. 19. So: $$ x - 3 = 4 \cdot x^{-x} $$ 20. This is transcendental; let's test values $x>3$: At $x=4$: Left $=4-3=1$, Right $=4 \cdot 4^{-4} = 4 \cdot \frac{1}{256} = \frac{1}{64} = 0.015625$; left > right. At $x=3.5$: Left $=0.5$, Right $=4 \cdot (3.5)^{-3.5} \approx 4 \cdot 0.032 =0.128$, left > right. At $x=3.1$: Left $=0.1$, Right $=4 \cdot (3.1)^{-3.1} \approx 4 \cdot 0.073=0.292$, left < right. 21. Since function crosses between 3.1 and 3.5, solution is near $3.3$. 22. Approximate $x \approx 3.3$. Final Answer: The solution is approximately $x \approx 3.3$.