# Honors Algebra 2: Unit 2 Review

• Class: Honors Algebra 2
• Author: Peter Atlas
• Text: Algebra and Trigonometry: Structure and Method, Brown

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1. Solve: $$\log_5 {x} + 3 \log_{25}{x} = 5$$
2. Solution

$$\displaystyle 25$$

3. Solve: $$\log_2{(x + 2)} + 2\log_4{(x - 2)} = 5$$
4. Solution

$$\displaystyle 6$$

5. Simplify: $$\log_{10}{11} \cdot \log_{11}{12} \cdot \log_{12}{13} \cdot \ldots \cdot \log_{998}{999} \cdot \log_{999}{1000}$$
6. Solution

$$\displaystyle 3$$

7. Solve: $$\log_{3}{(x + 3)} + \log_{3}{(2x + 1)} \gt \log_{3}{12}$$
8. Solution

$$\displaystyle x \gt 1$$

9. Simplify: $$\frac{\log_{c}{a^{\log_{a}{c}}} - \log_{c}{c^2}}{\left( \log_{b^2}{d^2} \right) \left( \log_{d}{b^6} \right)}$$
10. Solution

$$\displaystyle -\frac{1}{6}$$

11. Solve: $$\displaystyle e^{\ln{x^{\ln{4}}}} = 2$$
12. Solution

$$\displaystyle e^{\frac{2}{\ln{4}}}$$

13. Solve: $$\log_{2}{(x^2 + 2x + 1)} \le 2^{\log_{5}{25}}$$
14. Solution

$$\displaystyle x \in [-5, -1) \cup (-1, 3]$$

15. Simplify: $$\displaystyle \frac{e^{\ln{\left( \log_{7}{\left( \sqrt{2}^{\log_{2.5}{7}} \right)} \right)}}}{2\log_{4}{16} \cdot \frac{1}{\log_{3}{9}}} - \log_{\pi}{1}$$
16. Solution

$$\displaystyle \frac{1}{2}$$

17. Solve: $$\displaystyle \log_{3}{(2x + 3) \ge \log_{9}{x^2}} + \log_{27}{x^3}$$
18. Solution

$$\displaystyle x \in (0, 3)$$

19. Solve: $$\displaystyle \log_{4}{(x + 4) + \log_{16}{\left( x^2 - 8x + 16 \right)}} \le \log_{2}{\left( \frac{x^2 - 5x + 6}{x - 3} \right)}$$
20. Solution

$$\displaystyle x \le 5$$

21. Solve: $$\displaystyle \log_{7}{(x + 16)} + \log_{49}{\left( x^2 + 2x - 63\right)^2}$$
22. Solution

$$\displaystyle x = 8$$

23. Solve: $$\displaystyle \log_{2}{\left[ (2x - 6)(x + 2) \right]} - \log_{2}{(x^2 - x - 6)} \ge \log_{16}{(x + 6)}$$
24. Solution

$$\displaystyle x \in (-6, -2) \cup (3, 10]$$

25. Solve: $$\displaystyle \log_{2}{\sqrt{\log_{2}{3}}} = \log_{4}{( \log_{4}{x}) }$$
26. Solution

$$\displaystyle x = 9$$

27. Solve: $$\displaystyle \log_{3}{\left( x\sqrt{27} \right)} = \left( \log_{3}{x} \right)^{-1}$$
28. Solution

$$\displaystyle x = \left\{ \sqrt{3}, \frac{1}{9} \right\}$$