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Study Guides > College Algebra

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(12x)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(123x)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(x5)f\left(x\right)={\mathrm{log}}_{b}\left(x - 5\right) 12. g(x)=ln(3x)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(2x)f\left(x\right)=\mathrm{ln}\left(2-x\right) 17. f(x)=log(x37)f\left(x\right)=\mathrm{log}\left(x-\frac{3}{7}\right) 18. h(x)=log(3x4)+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(155x)+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(x1)+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(63x)+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(x1)+2=ln(x1)+2\mathrm{log}\left(x - 1\right)+2=\mathrm{ln}\left(x - 1\right)+2 52. log(2x3)+2=log(2x3)+5\mathrm{log}\left(2x - 3\right)+2=-\mathrm{log}\left(2x - 3\right)+5 53. ln(x2)=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(1x)=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 b1b\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+2x4)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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  • Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175..