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Guias de estudo > MTH 163, Precalculus

Section Exercises

1. The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each? 2. What type(s) of translation(s), if any, affect the range of a logarithmic function? 3. What type(s) of translation(s), if any, affect the domain of a logarithmic function? 4. Consider the general logarithmic function f(x)=logb(x)f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\\. Why can’t x be zero? 5. Does the graph of a general logarithmic function have a horizontal asymptote? Explain. For the following exercises, state the domain and range of the function. 6. f(x)=log3(x+4)f\left(x\right)={\mathrm{log}}_{3}\left(x+4\right)\\ 7. h(x)=ln(12−x)h\left(x\right)=\mathrm{ln}\left(\frac{1}{2}-x\right)\\ 8. g(x)=log5(2x+9)−2g\left(x\right)={\mathrm{log}}_{5}\left(2x+9\right)-2\\ 9. h(x)=ln(4x+17)−5h\left(x\right)=\mathrm{ln}\left(4x+17\right)-5\\ 10. f(x)=log2(12−3x)−3f\left(x\right)={\mathrm{log}}_{2}\left(12 - 3x\right)-3\\ For the following exercises, state the domain and the vertical asymptote of the function. 11. f(x)=logb(x−5)f\left(x\right)={\mathrm{log}}_{b}\left(x - 5\right)\\ 12. g(x)=ln(3−x)g\left(x\right)=\mathrm{ln}\left(3-x\right)\\ 13. f(x)=log(3x+1)f\left(x\right)=\mathrm{log}\left(3x+1\right)\\ 14. f(x)=3log(−x)+2f\left(x\right)=3\mathrm{log}\left(-x\right)+2\\ 15. g(x)=−ln(3x+9)−7g\left(x\right)=-\mathrm{ln}\left(3x+9\right)-7\\ For the following exercises, state the domain, vertical asymptote, and end behavior of the function. 16. f(x)=ln(2−x)f\left(x\right)=\mathrm{ln}\left(2-x\right)\\ 17. f(x)=log(x−37)f\left(x\right)=\mathrm{log}\left(x-\frac{3}{7}\right)\\ 18. h(x)=−log(3x−4)+3h\left(x\right)=-\mathrm{log}\left(3x - 4\right)+3\\ 19. g(x)=ln(2x+6)−5g\left(x\right)=\mathrm{ln}\left(2x+6\right)-5\\ 20. f(x)=log3(15−5x)+6f\left(x\right)={\mathrm{log}}_{3}\left(15 - 5x\right)+6\\ For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE. 21. h(x)=log4(x−1)+1h\left(x\right)={\mathrm{log}}_{4}\left(x - 1\right)+1\\ 22. f(x)=log(5x+10)+3f\left(x\right)=\mathrm{log}\left(5x+10\right)+3\\ 23. g(x)=ln(−x)−2g\left(x\right)=\mathrm{ln}\left(-x\right)-2\\ 24. f(x)=log2(x+2)−5f\left(x\right)={\mathrm{log}}_{2}\left(x+2\right)-5\\ 25. h(x)=3ln(x)−9h\left(x\right)=3\mathrm{ln}\left(x\right)-9\\ For the following exercises, match each function in the graph below with the letter corresponding to its graph. Graph of five logarithmic functions. 26. d(x)=log(x)d\left(x\right)=\mathrm{log}\left(x\right)\\ 27. f(x)=ln(x)f\left(x\right)=\mathrm{ln}\left(x\right)\\ 28. g(x)=log2(x)g\left(x\right)={\mathrm{log}}_{2}\left(x\right)\\ 29. h(x)=log5(x)h\left(x\right)={\mathrm{log}}_{5}\left(x\right)\\ 30. j(x)=log25(x)j\left(x\right)={\mathrm{log}}_{25}\left(x\right)\\ For the following exercises, match each function in the figure below with the letter corresponding to its graph. Graph of three logarithmic functions. 31. f(x)=log13(x)f\left(x\right)={\mathrm{log}}_{\frac{1}{3}}\left(x\right)\\ 32. g(x)=log2(x)g\left(x\right)={\mathrm{log}}_{2}\left(x\right)\\ 33. h(x)=log34(x)h\left(x\right)={\mathrm{log}}_{\frac{3}{4}}\left(x\right)\\ For the following exercises, sketch the graphs of each pair of functions on the same axis. 34. f(x)=log(x)f\left(x\right)=\mathrm{log}\left(x\right)\\ and g(x)=10xg\left(x\right)={10}^{x}\\ 35. f(x)=log(x)f\left(x\right)=\mathrm{log}\left(x\right)\\ and g(x)=log12(x)g\left(x\right)={\mathrm{log}}_{\frac{1}{2}}\left(x\right)\\ 36. f(x)=log4(x)f\left(x\right)={\mathrm{log}}_{4}\left(x\right)\\ and g(x)=ln(x)g\left(x\right)=\mathrm{ln}\left(x\right)\\ 37. f(x)=exf\left(x\right)={e}^{x}\\ and g(x)=ln(x)g\left(x\right)=\mathrm{ln}\left(x\right)\\ For the following exercises, match each function in the graph below with the letter corresponding to its graph. Graph of three logarithmic functions. 38. f(x)=log4(−x+2)f\left(x\right)={\mathrm{log}}_{4}\left(-x+2\right)\\ 39. g(x)=−log4(x+2)g\left(x\right)=-{\mathrm{log}}_{4}\left(x+2\right)\\ 40. h(x)=log4(x+2)h\left(x\right)={\mathrm{log}}_{4}\left(x+2\right)\\ For the following exercises, sketch the graph of the indicated function. 41. f(x)=log2(x+2)f\left(x\right)={\mathrm{log}}_{2}\left(x+2\right)\\ 42. f(x)=2log(x)f\left(x\right)=2\mathrm{log}\left(x\right)\\ 43. f(x)=ln(−x)f\left(x\right)=\mathrm{ln}\left(-x\right)\\ 44. g(x)=log(4x+16)+4g\left(x\right)=\mathrm{log}\left(4x+16\right)+4\\ 45. g(x)=log(6−3x)+1g\left(x\right)=\mathrm{log}\left(6 - 3x\right)+1\\ 46. h(x)=−12ln(x+1)−3h\left(x\right)=-\frac{1}{2}\mathrm{ln}\left(x+1\right)-3\\ For the following exercises, write a logarithmic equation corresponding to the graph shown. 47. Use y=log2(x)y={\mathrm{log}}_{2}\left(x\right)\\ as the parent function. The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1. 48. Use f(x)=log3(x)f\left(x\right)={\mathrm{log}}_{3}\left(x\right)\\ as the parent function. The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4. 49. Use f(x)=log4(x)f\left(x\right)={\mathrm{log}}_{4}\left(x\right)\\ as the parent function. The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2. 50. Use f(x)=log5(x)f\left(x\right)={\mathrm{log}}_{5}\left(x\right)\\ as the parent function. The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5. For the following exercises, use a graphing calculator to find approximate solutions to each equation. 51. log(x−1)+2=ln(x−1)+2\mathrm{log}\left(x - 1\right)+2=\mathrm{ln}\left(x - 1\right)+2\\ 52. log(2x−3)+2=−log(2x−3)+5\mathrm{log}\left(2x - 3\right)+2=-\mathrm{log}\left(2x - 3\right)+5\\ 53. ln(x−2)=−ln(x+1)\mathrm{ln}\left(x - 2\right)=-\mathrm{ln}\left(x+1\right)\\ 54. 2ln(5x+1)=12ln(−5x)+12\mathrm{ln}\left(5x+1\right)=\frac{1}{2}\mathrm{ln}\left(-5x\right)+1\\ 55. 13log(1−x)=log(x+1)+13\frac{1}{3}\mathrm{log}\left(1-x\right)=\mathrm{log}\left(x+1\right)+\frac{1}{3}\\ 56. Let b be any positive real number such that b≠1b\ne 1\\. What must logb1{\mathrm{log}}_{b}1\\ be equal to? Verify the result. 57. Explore and discuss 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)\\. Make a conjecture based on the result. 58. Prove the conjecture made in the previous exercise. 59. What is the domain of the function f(x)=ln(x+2x−4)f\left(x\right)=\mathrm{ln}\left(\frac{x+2}{x - 4}\right)\\? Discuss the result. 60. Use properties of exponents to find the x-intercepts of the function f(x)=log(x2+4x+4)f\left(x\right)=\mathrm{log}\left({x}^{2}+4x+4\right)\\ algebraically. Show the steps for solving, and then verify the result by graphing the function.

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