1. **State the problem:** Solve the logarithmic equation $$\log_2(5 + 2x) - \log_2(4 - x) = 3$$. 2. **Recall the logarithm property:** The difference of logarithms with the same base can be written as the logarithm of a quotient: $$\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)$$. 3. **Apply the property:** $$\log_2\left(\frac{5 + 2x}{4 - x}\right) = 3$$. 4. **Rewrite the logarithmic equation in exponential form:** $$\frac{5 + 2x}{4 - x} = 2^3$$. 5. **Calculate the right side:** $$2^3 = 8$$. 6. **Set up the equation:** $$\frac{5 + 2x}{4 - x} = 8$$. 7. **Cross multiply:** $$5 + 2x = 8(4 - x)$$. 8. **Distribute:** $$5 + 2x = 32 - 8x$$. 9. **Bring all terms to one side:** $$2x + 8x = 32 - 5$$ $$10x = 27$$. 10. **Solve for x:** $$x = \frac{27}{10} = 2.7$$. 11. **Check the domain:** - Argument of first log: $5 + 2x = 5 + 2(2.7) = 5 + 5.4 = 10.4 > 0$ - Argument of second log: $4 - x = 4 - 2.7 = 1.3 > 0$ Both arguments are positive, so $x=2.7$ is valid. **Final answer:** $$x = 2.7$$