Exponential Functions

1. exponential; the population decreases by a proportional rate.
2. not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function.
3. The forest represented by the function [latex]B(t)=82(1.029)^t[/latex].
4. After [latex]t=20[/latex] years, forest A will have [latex]43[/latex] more trees than forest B.
5. exponential growth: The growth factor, [latex]1.06[/latex], is greater than [latex]1[/latex].
6. exponential decay: The decay factor, [latex]0.97[/latex], is between [latex]0[/latex] and [latex]1[/latex].
7. [latex]f(x)=2000(0.1)^x[/latex]
8. [latex]f(x)=(\dfrac{1}{6})^{-\frac{3}{5}}(\dfrac{1}{6})^{\frac{x}{5}}\approx 2.93(0.699)^x[/latex]
9. Linear
10. Neither
11. Linear
12. [latex]f(-1)=-4[/latex]
13. [latex]f(-1)\approx -0.2707[/latex]
14. [latex]f(3)\approx 483.8146[/latex]
15. [latex]g(x)=4(3)^{-x}[/latex]; [latex]y[/latex]-intercept: [latex](0,4)[/latex]; Domain: all real numbers; Range: all real numbers greater than [latex]0[/latex].
16. [latex]g(x)=-10^x+7[/latex]; [latex]y[/latex]-intercept: [latex](0,6)[/latex]; Domain: all real numbers; Range: all real numbers less than [latex]7[/latex].
17. [latex]g(x)=2(\dfrac{1}{4})^x[/latex]; [latex]y[/latex]-intercept: [latex](0,2)[/latex]; Domain: all real numbers; Range: all real numbers greater than [latex]0[/latex].
18. [latex]y[/latex]-intercept: [latex](0,-2)[/latex]
    
19. 
20. B
21. A
22. E
23. D
24. C
25. [latex]f(x)=4^x-3[/latex]
26. [latex]f(x)=4^{x-5}[/latex]
27. [latex]f(x)=4^{-x}[/latex]
28. [latex]y=-2^x+3[/latex]
29. [latex]y=-2(3)^x+7[/latex]

Applications of Exponential Functions

1. [latex]$10,250[/latex]
2. [latex]$13,268.58[/latex]
3. [latex]P=A(t)\cdot(1+\dfrac{r}{n})^{-nt}[/latex]
4. [latex]$4,572.56[/latex]
5. [latex]4 \%[/latex]
6. continuous growth; the growth rate is greater than [latex]0[/latex].
7. continuous decay; the growth rate is less than [latex]0[/latex].
8. [latex]47,622[/latex] fox
9. [latex]1.39 \%[/latex]; [latex]$155,368.09[/latex]
10. [latex]$35,838.76[/latex]
11. [latex]$82,247.78[/latex]; [latex]$449.75[/latex]

Logarithmic Functions

1. [latex]a^c = b[/latex]
2. [latex]x^y = 64[/latex]
3. [latex]15^b = a[/latex]
4. [latex]\log_c(k) = d[/latex]
5. [latex]\log_{19}y = x[/latex]
6. [latex]\log_n(103) = 4[/latex]
7. [latex]\log_y(\dfrac{39}{100}) = x[/latex]
8. [latex]x = 2^{-3} = \dfrac{1}{8}[/latex]
9. [latex]x = 3^3 = 27[/latex]
10. [latex]x = 9^{\frac{1}{2}} = 3[/latex]
11. [latex]32[/latex]
12. [latex]1.06[/latex]
13. [latex]14.125[/latex]
14. [latex]\dfrac{1}{2}[/latex]
15. [latex]4[/latex]
16. [latex]-3[/latex]
17. [latex]-12[/latex]
18. [latex]0[/latex]
19. [latex]10[/latex]

Logarithmic Function Graphs and Characteristics

1. Domain: [latex](-\infty,\dfrac{1}{2})[/latex]; Range: [latex](-\infty,\infty)[/latex]
2. Domain: [latex](-\dfrac{17}{4},\infty)[/latex]; Range: [latex](-\infty,\infty)[/latex]
3. Domain: [latex](5,\infty)[/latex]; Vertical asymptote: [latex]x=5[/latex]
4. Domain: [latex](-\dfrac{1}{3},\infty)[/latex]; Vertical asymptote: [latex]x=-\dfrac{1}{3}[/latex]
5. Domain: [latex](-3,\infty)[/latex]; Vertical asymptote: [latex]x=-3[/latex]
6. Domain: [latex](1,\infty)[/latex]; Range: [latex](-\infty,\infty)[/latex]; Vertical asymptote: [latex]x=1[/latex]; [latex]x[/latex]-intercept: [latex](\dfrac{5}{4},0)[/latex]; [latex]y[/latex]-intercept: DNE
7. Domain: [latex](-\infty,0)[/latex]; Range: [latex](-\infty,\infty)[/latex]; Vertical asymptote: [latex]x=0[/latex]; [latex]x[/latex]-intercept: [latex](-e^2,0)[/latex]; [latex]y[/latex]-intercept: DNE
8. Domain: [latex](0,\infty)[/latex]; Range: [latex](-\infty,\infty)[/latex]; Vertical asymptote: [latex]x=0[/latex]; [latex]x[/latex]-intercept: [latex](e^3,0)[/latex]; [latex]y[/latex]-intercept: DNE
9. B
10. C
11. B
12. C
13. 
14. 
15. 
16. 
17. 
18. [latex]f(x)=\log_2(-x-1)[/latex]
19. [latex]f(x)=3\log_4(x+2)[/latex]