4.4E: Logarithmic Properties (Exercises)

Section 4.4 Exercises

Simplify to a single logarithm, using logarithm properties. \[1. \log _{3} \left(28\right)-\log _{3} \left(7\right) 2. \log _{3} \left(32\right)-\log _{3} \left(4\right)\] \[3. -\log _{3} \left(\frac{1}{7} \right) 4. -\log _{4} \left(\frac{1}{5} \right)\] \[5. \log _{3} \left(\frac{1}{10} \right)+\log _{3} \left(50\right) 6. \log _{4} \left(3\right)+\log _{4} (7)\] \[7. \frac{1}{3} \log _{7} \left(8\right) 8. \frac{1}{2} \log _{5} \left(36\right)\] \[9. \log \left(2x^{4} \right)+\log \left(3x^{5} \right) 10. \ln \left(4x^{2} \right)+\ln \left(3x^{3} \right)\] \[11. \ln \left(6x^{9} \right)-\ln \left(3x^{2} \right) 12. \log \left(12x^{4} \right)-\log \left(4x\right)\] \[13. 2\log \left(x\right)+3\log \left(x+1\right) 14. 3\log \left(x\right)+2\log \left(x^{2} \right)\] \[15. \log \left(x\right)-\frac{1}{2} \log \left(y\right)+3\log \left(z\right) 16. 2\log \left(x\right)+\frac{1}{3} \log \left(y\right)-\log \left(z\right)\]

Use logarithm properties to expand each expression. \[17. \log \left(\frac{x^{15} y^{13} }{z^{19} } \right) 18. \log \left(\frac{a^{2} b^{3} }{c^{5} } \right)\] \[19. \ln \left(\frac{a^{-2} }{b^{-4} c^{5} } \right) 20. \ln \left(\frac{a^{-2} b^{3} }{c^{-5} } \right)\] \[21. \log \left(\sqrt{x^{3} y^{-4} } \right) 22. \log \left(\sqrt{x^{-3} y^{2} } \right)\] \[23. \ln \left(y\sqrt{\frac{y}{1-y} } \right) 24. \ln \left(\frac{x}{\sqrt{1-x^{2} } } \right)\] \[25. \log \left(x^{2} y^{3} \sqrt[{3}]{x^{2} y^{5} } \right) 26. \log \left(x^{3} y^{4} \sqrt[{7}]{x^{3} y^{9} } \right)\]

Solve each equation for the variable. \[27. 4^{4x-7} =3^{9x-6} 28. 2^{2x-5} =7^{3x-7}\] \[29. 17\left(1.14\right)^{x} =19\left(1.16\right)^{x} 30. 20\left(1.07\right)^{x} =8\left(1.13\right)^{x}\] \[31. 5e^{0.12t} =10e^{0.08t} 32. 3e^{0.09t} =e^{0.14t}\] \[33. \log _{2} \left(7x+6\right)=3 34. \log _{3} (2x+4)=2\] \[35. 2\ln \left(3{\rm x}\right)+3=1 36. 4\ln \left(5x\right)+5=2\] \[37. \log \left(x^{3} \right)=2 38. \log \left(x^{5} \right)=3\] \[39. \log \left(x\right)+\log \left(x+3\right)=3 40. \log \left(x+4\right)+\log \left(x\right)=9\] \[41. \log \left(x+4\right)-\log \left(x+3\right)=1 42. \log \left(x+5\right)-\log \left(x+2\right)=2\] \[43. \log _{6} \left(x^{2} \right)-\log _{6} (x+1)=1 44. \log _{3} (x^{2} )-\log _{3} (x+2)=5\] \[45. \log \left(x+12\right)=\log \left(x\right)+\log \left(12\right) 46. \log \left(x+15\right)=\log \left(x\right)+\log \left(15\right)\] \[47. \ln \left(x\right)+\ln \left(x-3\right)=\ln \left(7x\right) 48. \ln \left(x\right)+\ln \left(x-6\right)=\ln \left(6x\right)\]

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Section 4.5 Graphs of Logarithmic Functions