1. \[ -6 \]
    
2. \[ 4 \]
    
3. \[ -4 \]
    
4. \[ -\frac{1}{2} \]
    
5. \[ 3 \]
    
6. \[ 9 \]
    
7. \[ \sqrt{3} \]
    
8. \[ \frac{8}{9} \]
    
9. \[ 625 \]
    
10. \[ 8, \, -2 \]
     
11. \[ 5 \]
     
12. \[ \mathrm{e}^{-\frac{a}{3}} \]
     
13. \[ \mathrm{e}^{ -1 + \frac{k}{20} } \]
     
14. \[ \mathrm{e}^2 \]
     
15. \[ \mathrm{e}^{ \mathrm{e}^{-4} } \]
     
16. \[ \frac { \ln \left( \frac{14}{3} \right) }{ -1.2 } \approx 1.2837 \]
     
17. \[ 1 + \frac{3}{\ln 3} \approx 3.73 \]
     
18. \[ \frac{6} { (3 \log_2 3) } \approx 1.3086 \]
     
19. \[ \frac{ 8 }{ (4 \log_2 3 – 1 ) }\approx 1.498 \]
     
20. \(y = \ln (x – 2)\)

    

21. \(y = \ln (-x)\)

    

22. \(y = \ln (x + 3)\)

    

23. \(y = 4 – \ln x\)

    

24. \(y = 4 – \ln (x + 3)\)

    

25. \(y = 2 – \ln \mid x \mid\)

    

26. \(y = 3 + \log x\)

    

27. \(y = 3 + \log (x + 3)\)

    

28. \[ 2 \log a + \log b – \log c \]
     
29. \[ \frac{1}{2} \log b – 2 \log a – 3 \log c \]
     
30. \[ – \ln a + \frac{3}{2} \ln c – \frac{1}{2} \ln b \]
     
31. \[ \frac{1}{5} ( 2 \ln a – \ln b – 4 \ln c ) \]
     
32. \[ \ln \frac{ x^3 y }{z^2} \]
     
33. \[ \log \frac{ a^2 b } {(zx)^3} \]
     
34. \[ \ln \frac{ \sqrt[4]{a^3} b^3 }{ \sqrt{c^3} } \]
     
35.  
    1. \(y = 5 \mathrm{e}^{ (0.5 \ln 3)t }\)
    2. \(y = 6 \mathrm{e}^{ ( \ln ( 1.04 ) ) t }\)

In exercises 1 through 8, find the value of the expression without using tables or calculator.

1. \[ \log_2 \left( \frac{1}{64} \right) \]
2. \[ \log_{1/2} \left( \frac{1}{16} \right) \]
3. \[ \log_{1/3} (81) \]
4. \[ \log_{100} (0.1) \]
5. \[ \mathrm{e}^{\ln 3} \]
6. \[ \mathrm{e}^{2 \ln 3} \]
7. \[ \mathrm{e}^{(\ln 3)/2} \]
8. \[ \mathrm{e}^{3 \ln {2} – 2 \ln 3} \]

In exercises 9 through 19, solve the given equation.

9. \[ \log_x (25) = \frac{1}{2} \]
10. \[ \log_4 \left( x^2 – 6x \right) = 2 \]
11. \[ \log x + \log (2x – 8) = 1 \]
12. \[ -3 \ln x = a \]
13. \[ \frac{k}{20} – \ln x = 1 \]
14. \[ 4 \ln x = \frac{1}{2} \ln x + 7 \]
15. \[ 3 \ln (\ln x) = -12 \]
16. \[ 3 \mathrm{e}^{-1.2x} = 14 \]
17. \[ 3^{x-1} = \mathrm{e^3} \]
18. \[ 3^x 2^{3x} = 64 \]
19. \[ \left( 3^x \right)^2 = 16\sqrt{2^x} \]

In exercises 20 through 27, use translation and reflection techniques to sketch the graph of the given functions.

20. \[ y = \ln (x – 2 ) \]
21. \[ y=\ln (-x ) \]
22. \[ y = \ln (x + 3) \]
23. \[ y = 4 – \ln x \]
24. \[ y = 4 – \ln (x + 3) \]
25. \[ y = 2 – \ln {\mid x \mid} \]
26. \[ y = 3 + \log x \]
27. \[ y = 3 + \log (x + 3) \]

In exercises 28 through 31, write the expression in terms of the logarithms of (\boldsymbol{a}\), \(\boldsymbol{b}\) and \(\boldsymbol{c}\).

28. \[ \log { \cfrac{a^2 b}{c} } \]
29. \[ \log { \cfrac{\sqrt{b}}{a^2 c^3} } \]
30. \[ \ln \left( \cfrac{1}{a} \sqrt{ \cfrac{c^3}{b} } \right) \]
31. \[ \ln \sqrt[5]{ \cfrac{a^2}{b c^4} } \]

In the exercises 32, 33 and 34, rewrite the expression using only one logarithm of coefficient 1.

32. \[ 3\ln x + \ln y – 2 \ln z \]
33. \[ 2 \log a + \log b – 3( \log z + \log x) \]
34. \[ \frac{3}{4} \ln a + 3 \ln b -\frac{3}{2} \ln c \]

35. Express each of the following functions with the form \(y=A\mathrm{e}^{kt}\):

    1. \[ y = (5)3^{0.5t} \]
    2. \[ y = 6(1.04)^t \]