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4.4E: Logarithmic Properties (Exercises)

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    section 4.4 exercise

    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. \(-\log _{3} \left(\dfrac{1}{7} \right)\)

    4. \(-\log _{4} \left(\dfrac{1}{5} \right)\)

    5. \(\log _{3} \left(\dfrac{1}{10} \right)+\log _{3} \left(50\right)\)

    6. \(\log _{4} \left(3\right)+\log _{4} (7)\)

    7. \(\dfrac{1}{3} \log _{7} \left(8\right)\)

    8. \(\dfrac{1}{2} \log _{5} \left(36\right)\)

    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. \(\log \left(x\right)-\dfrac{1}{2} \log \left(y\right)+3\log \left(z\right)\)

    16. \(2\log \left(x\right)+\dfrac{1}{3} \log \left(y\right)-\log \left(z\right)\)

    Use logarithm properties to expand each expression.

    17. \(\log \left(\dfrac{x^{15} y^{13} }{z^{19} } \right)\)

    18. \(\log \left(\dfrac{a^{2} b^{3} }{c^{5} } \right)\)

    19. \(\ln \left(\dfrac{a^{-2} }{b^{-4} c^{5} } \right)\)

    20. \(\ln \left(\dfrac{a^{-2} b^{3} }{c^{-5} } \right)\)

    21. \(\log \left(\sqrt{x^{3} y^{-4} } \right)\)

    22. \(\log \left(\sqrt{x^{-3} y^{2} } \right)\)

    23. \(\ln \left(y\sqrt{\dfrac{y}{1-y} } \right)\)

    24. \(\ln \left(\dfrac{x}{\sqrt{1-x^{2} } } \right)\)

    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. \(\ln \left(x\right)+\ln \left(x-6\right)=\ln \left(6x\right)\)

    Answer

    1. \(\text{log}_3 (4)\)

    3. \(\text{log}_3 (7)\)

    5. \(\text{log}_3 (5)\)

    7. \(\text{log}_7 (2)\)

    9. \(\text{log} (6x^9)\)

    11. \(\text{ln} (2x^7)\)

    13. \(\text{log}(x^2 (x + 1)^3)\)

    15. \(\text{log} (\dfrac{xz^3}{\sqrt{y}})\)

    17. \(15\text{log}(x) + 13 \text{log}(y) - 19 \text{log}(z)\)

    19. \(-2\text{ln} (a) + 4\text{ln}(b) - 5 \text{ln}(c)\)

    21. \(\dfrac{3}{2} \text{log}(x) - 2 \text{log}(y)\)

    23. \(\text{ln}(y) + \dfrac{1}{2} (\text{ln} (y) - \text{ln} (1 - y))\)

    25. \(\dfrac{8}{3} \text{log} (x) + \dfrac{14}{3} \text{log} (y)\)

    27. \(x \approx -0.717\)

    29. \(x \approx -6.395\)

    31. \(t \approx 17.329\)

    33. \(x = \dfrac{2}{7}\)

    35. \(x \approx 0.123\)

    37. \(x \approx 4.642\)

    39. \(x \approx 30.158\)

    41. \(x \approx -2.889\)

    43. \(x \approx 6.873\) or \(x \approx -0.873\)

    45. \(x = \dfrac{12}{11} \approx 1.091\)

    47. \(x = 10\)

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