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Study Guides > MATH 1314: College Algebra

Solutions

Solutions to Try Its

1. (2,)\left(2,\infty \right)\\ 2. (5,)\left(5,\infty \right)\\ 3. The domain is (0,)\left(0,\infty \right)\\, the range is (,)\left(-\infty ,\infty \right)\\, and the vertical asymptote is = 0. Graph of f(x)=log_(1/5)(x) with labeled points at (1/5, 1) and (1, 0). The y-axis is the asymptote. 4. The domain is (4,)\left(-4,\infty \right)\\, the range (,)\left(-\infty ,\infty \right)\\, and the asymptote = –4. Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1). 5. The domain is (0,)\left(0,\infty \right)\\, the range is (,)\left(-\infty ,\infty \right)\\, and the vertical asymptote is = 0. Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1). 6. The domain is (0,)\left(0,\infty \right)\\, the range is (,)\left(-\infty ,\infty \right)\\, and the vertical asymptote is = 0. Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1). 7. The domain is (2,)\left(2,\infty \right)\\, the range is (,)\left(-\infty ,\infty \right)\\, and the vertical asymptote is = 2.
Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.
8. The domain is (,0)\left(-\infty ,0\right)\\, the range is (,)\left(-\infty ,\infty \right)\\, and the vertical asymptote is = 0. Graph of f(x)=-log(-x) with an asymptote at x=0.

9. x3.049x\approx 3.049\\

10. = 1 11. f(x)=2ln(x+3)1f\left(x\right)=2\mathrm{ln}\left(x+3\right)-1\\

Solutions to Odd-Numbered Exercises

1. Since the functions are inverses, their graphs are mirror images about the line = x. So for every point (a,b)\left(a,b\right)\\ on the graph of a logarithmic function, there is a corresponding point (b,a)\left(b,a\right)\\ on the graph of its inverse exponential function. 3. Shifting the function right or left and reflecting the function about the y-axis will affect its domain. 5. No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers. 7. Domain: (,12)\left(-\infty ,\frac{1}{2}\right)\\; Range: (,)\left(-\infty ,\infty \right)\\ 9. Domain: (174,)\left(-\frac{17}{4},\infty \right)\\; Range: (,)\left(-\infty ,\infty \right)\\ 11. Domain: (5,)\left(5,\infty \right)\\; Vertical asymptote: = 5 13. Domain: (13,)\left(-\frac{1}{3},\infty \right)\\; Vertical asymptote: x=13x=-\frac{1}{3}\\ 15. Domain: (3,)\left(-3,\infty \right)\\; Vertical asymptote: = –3 17. Domain: (37,)\left(\frac{3}{7},\infty \right)\\; Vertical asymptote: x=37x=\frac{3}{7}\\ ; End behavior: as x(37)+,f(x)x\to {\left(\frac{3}{7}\right)}^{+},f\left(x\right)\to -\infty \\ and as x,f(x)x\to \infty ,f\left(x\right)\to \infty \\ 19. Domain: (3,)\left(-3,\infty \right)\\ ; Vertical asymptote: = –3; End behavior: as x3+x\to -{3}^{+}\\f(x)f\left(x\right)\to -\infty \\ and as xx\to \infty \\f(x)f\left(x\right)\to \infty \\ 21. Domain: (1,)\left(1,\infty \right)\\; Range: (,)\left(-\infty ,\infty \right)\\; Vertical asymptote: = 1; x-intercept: (54,0)\left(\frac{5}{4},0\right)\\; y-intercept: DNE 23. Domain: (,0)\left(-\infty ,0\right)\\; Range: (,)\left(-\infty ,\infty \right)\\; Vertical asymptote: = 0; x-intercept: (e2,0)\left(-{e}^{2},0\right)\\; y-intercept: DNE 25. Domain: (0,)\left(0,\infty \right)\\; Range: (,)\left(-\infty ,\infty \right)\\; Vertical asymptote: = 0; x-intercept: (e3,0)\left({e}^{3},0\right); y-intercept: DNE 27. B 29. C 31. B 33. C 35. Graph of two functions, g(x) = log_(1/2)(x) in orange and f(x)=log(x) in blue. 37. Graph of two functions, g(x) = ln(1/2)(x) in orange and f(x)=e^(x) in blue. 39. C 41. Graph of f(x)=log_2(x+2). 43. Graph of f(x)=ln(-x). 45. Graph of g(x)=log(6-3x)+1. 47. f(x)=log2((x1))f\left(x\right)={\mathrm{log}}_{2}\left(-\left(x - 1\right)\right)\\ 49. f(x)=3log4(x+2)f\left(x\right)=3{\mathrm{log}}_{4}\left(x+2\right)\\ 51. = 2 53. x2.303x\approx \text{2}\text{.303}\\ 55. x0.472x\approx -0.472\\ 57. The graphs of f(x)=log12(x)f\left(x\right)={\mathrm{log}}_{\frac{1}{2}}\left(x\right)\\ and g(x)=log2(x)g\left(x\right)=-{\mathrm{log}}_{2}\left(x\right)\\ appear to be the same; Conjecture: for any positive base b1b\ne 1\\, logb(x)=log1b(x){\mathrm{log}}_{b}\left(x\right)=-{\mathrm{log}}_{\frac{1}{b}}\left(x\right)\\. 59. Recall that the argument of a logarithmic function must be positive, so we determine where x+2x4>0\frac{x+2}{x - 4}>0\\ . From the graph of the function f(x)=x+2x4f\left(x\right)=\frac{x+2}{x - 4}\\, note that the graph lies above the x-axis on the interval (,2)\left(-\infty ,-2\right)\\ and again to the right of the vertical asymptote, that is (4,)\left(4,\infty \right)\\. Therefore, the domain is (,2)(4,)\left(-\infty ,-2\right)\cup \left(4,\infty \right)\\.

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