{"id":133,"date":"2026-01-12T15:50:05","date_gmt":"2026-01-12T15:50:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=133"},"modified":"2026-01-12T15:50:05","modified_gmt":"2026-01-12T15:50:05","slug":"exponential-and-logarithmic-equations-and-models-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/exponential-and-logarithmic-equations-and-models-get-stronger-answer-key\/","title":{"raw":"Exponential and Logarithmic Equations and Models: Get Stronger Answer Key","rendered":"Exponential and Logarithmic Equations and Models: Get Stronger Answer Key"},"content":{"raw":"

Logarithmic Properties<\/h2>\r\n
    \r\n \t
  1. [latex]\\log_b(2)+\\log_b(7)+\\log_b(x)+\\log_b(y)[\/latex]<\/li>\r\n \t
  2. [latex]\\log_b(13)-\\log_b(17)[\/latex]<\/li>\r\n \t
  3. [latex]-k\\ln(4)[\/latex]<\/li>\r\n \t
  4. [latex]\\ln(7xy)[\/latex]<\/li>\r\n \t
  5. [latex]\\log_b(4)[\/latex]<\/li>\r\n \t
  6. [latex]\\log_b(7)[\/latex]<\/li>\r\n \t
  7. [latex]15\\log(x)+13\\log(y)-19\\log(z)[\/latex]<\/li>\r\n \t
  8. [latex]\\dfrac{3}{2}\\log(x)-2\\log(y)[\/latex]<\/li>\r\n \t
  9. [latex]\\dfrac{8}{3}\\log(x)+\\dfrac{14}{3}\\log(y)[\/latex]<\/li>\r\n \t
  10. [latex]\\ln(2x^7)[\/latex]<\/li>\r\n \t
  11. [latex]\\log(\\dfrac{xz^3}{\\sqrt{y}})[\/latex]<\/li>\r\n \t
  12. [latex]\\log_7(15)=\\dfrac{\\ln(15)}{\\ln(7)}[\/latex]<\/li>\r\n \t
  13. [latex]\\log_{11}(5)=\\dfrac{\\log_5(5)}{\\log_5(11)}=\\dfrac{1}{b}[\/latex]<\/li>\r\n \t
  14. [latex]\\log_{11}(\\dfrac{6}{11})=\\dfrac{\\log_5(\\dfrac{6}{11})}{\\log_5(11)}=\\dfrac{\\log_5(6)-\\log_5(11)}{\\log_5(11)}=\\dfrac{a-b}{b}=\\dfrac{a}{b}-1[\/latex]<\/li>\r\n \t
  15. [latex]3[\/latex]<\/li>\r\n \t
  16. [latex]2.81359[\/latex]<\/li>\r\n \t
  17. [latex]0.93913[\/latex]<\/li>\r\n \t
  18. [latex]-2.23266[\/latex]<\/li>\r\n<\/ol>\r\n

    Exponential and Logarithmic Equations and Models<\/h2>\r\n
      \r\n \t
    1. [latex]x=-\\dfrac{1}{3}[\/latex]<\/li>\r\n \t
    2. [latex]n=-1[\/latex]<\/li>\r\n \t
    3. [latex]b=\\dfrac{6}{5}[\/latex]<\/li>\r\n \t
    4. [latex]x=10[\/latex]<\/li>\r\n \t
    5. No solution<\/li>\r\n \t
    6. [latex]p=\\log(\\dfrac{17}{8})-7[\/latex]<\/li>\r\n \t
    7. [latex]k=-\\dfrac{\\ln(38)}{3}[\/latex]<\/li>\r\n \t
    8. [latex]x=\\dfrac{\\ln(\\dfrac{38}{3})-8}{9}[\/latex]<\/li>\r\n \t
    9. [latex]x=\\ln(12)[\/latex]<\/li>\r\n \t
    10. [latex]10^{-2}=\\dfrac{1}{100}[\/latex]<\/li>\r\n \t
    11. [latex]n=49[\/latex]<\/li>\r\n \t
    12. [latex]k=\\dfrac{1}{36}[\/latex]<\/li>\r\n \t
    13. [latex]x=\\dfrac{9-e}{8}[\/latex]<\/li>\r\n \t
    14. [latex]n=1[\/latex]<\/li>\r\n \t
    15. No solution<\/li>\r\n \t
    16. No solution<\/li>\r\n \t
    17. [latex]x=\\pm\\dfrac{10}{3}[\/latex]<\/li>\r\n \t
    18. [latex]x=10[\/latex]<\/li>\r\n \t
    19. [latex]x=0[\/latex]<\/li>\r\n \t
    20. [latex]x=\\dfrac{3}{4}[\/latex]<\/li>\r\n<\/ol>\r\n

      Exponential and Logarithmic Models<\/h2>\r\n
        \r\n \t
      1. [latex]f(0)\\approx 16.7[\/latex]; The amount initially present is about [latex]16.7[\/latex] units.<\/li>\r\n \t
      2. [latex]150[\/latex]<\/li>\r\n \t
      3. exponential; [latex]f(x)=1.2^x[\/latex]<\/li>\r\n \t
      4. logarithmic\r\n\"Scatter<\/li>\r\n \t
      5. logarithmic\r\n\"Scatter<\/li>\r\n \t
      6. \"Graph<\/li>\r\n \t
      7. about [latex]1.4[\/latex] years<\/li>\r\n \t
      8. about [latex]7.3[\/latex] years<\/li>\r\n \t
      9. [latex]A=125e^{(-0.3567t)}[\/latex]; [latex]A\\approx 43[\/latex] mg<\/li>\r\n \t
      10. about [latex]60[\/latex] days<\/li>\r\n \t
      11. [latex]A(t)=250e^{(-0.00822t)}[\/latex]; half-life: about [latex]84[\/latex] minutes<\/li>\r\n \t
      12. [latex]r\\approx -0.0667[\/latex], So the hourly decay rate is about [latex]6.67 \\%[\/latex]<\/li>\r\n \t
      13. [latex]f(t)=1350e^{(0.03466t)}[\/latex]; after [latex]3[\/latex] hours: [latex]P(180)\\approx 691,200[\/latex]<\/li>\r\n \t
      14. [latex]f(t)=256e^{(0.068110t)}[\/latex]; doubling time: about [latex]10[\/latex] minutes<\/li>\r\n \t
      15. about [latex]88[\/latex] minutes<\/li>\r\n \t
      16. [latex]T(t)=90e^{(-0.008377t)}+75[\/latex], where [latex]t[\/latex] is in minutes.<\/li>\r\n \t
      17. about [latex]113[\/latex] minutes<\/li>\r\n<\/ol>","rendered":"

        Logarithmic Properties<\/h2>\n
          \n
        1. [latex]\\log_b(2)+\\log_b(7)+\\log_b(x)+\\log_b(y)[\/latex]<\/li>\n
        2. [latex]\\log_b(13)-\\log_b(17)[\/latex]<\/li>\n
        3. [latex]-k\\ln(4)[\/latex]<\/li>\n
        4. [latex]\\ln(7xy)[\/latex]<\/li>\n
        5. [latex]\\log_b(4)[\/latex]<\/li>\n
        6. [latex]\\log_b(7)[\/latex]<\/li>\n
        7. [latex]15\\log(x)+13\\log(y)-19\\log(z)[\/latex]<\/li>\n
        8. [latex]\\dfrac{3}{2}\\log(x)-2\\log(y)[\/latex]<\/li>\n
        9. [latex]\\dfrac{8}{3}\\log(x)+\\dfrac{14}{3}\\log(y)[\/latex]<\/li>\n
        10. [latex]\\ln(2x^7)[\/latex]<\/li>\n
        11. [latex]\\log(\\dfrac{xz^3}{\\sqrt{y}})[\/latex]<\/li>\n
        12. [latex]\\log_7(15)=\\dfrac{\\ln(15)}{\\ln(7)}[\/latex]<\/li>\n
        13. [latex]\\log_{11}(5)=\\dfrac{\\log_5(5)}{\\log_5(11)}=\\dfrac{1}{b}[\/latex]<\/li>\n
        14. [latex]\\log_{11}(\\dfrac{6}{11})=\\dfrac{\\log_5(\\dfrac{6}{11})}{\\log_5(11)}=\\dfrac{\\log_5(6)-\\log_5(11)}{\\log_5(11)}=\\dfrac{a-b}{b}=\\dfrac{a}{b}-1[\/latex]<\/li>\n
        15. [latex]3[\/latex]<\/li>\n
        16. [latex]2.81359[\/latex]<\/li>\n
        17. [latex]0.93913[\/latex]<\/li>\n
        18. [latex]-2.23266[\/latex]<\/li>\n<\/ol>\n

          Exponential and Logarithmic Equations and Models<\/h2>\n
            \n
          1. [latex]x=-\\dfrac{1}{3}[\/latex]<\/li>\n
          2. [latex]n=-1[\/latex]<\/li>\n
          3. [latex]b=\\dfrac{6}{5}[\/latex]<\/li>\n
          4. [latex]x=10[\/latex]<\/li>\n
          5. No solution<\/li>\n
          6. [latex]p=\\log(\\dfrac{17}{8})-7[\/latex]<\/li>\n
          7. [latex]k=-\\dfrac{\\ln(38)}{3}[\/latex]<\/li>\n
          8. [latex]x=\\dfrac{\\ln(\\dfrac{38}{3})-8}{9}[\/latex]<\/li>\n
          9. [latex]x=\\ln(12)[\/latex]<\/li>\n
          10. [latex]10^{-2}=\\dfrac{1}{100}[\/latex]<\/li>\n
          11. [latex]n=49[\/latex]<\/li>\n
          12. [latex]k=\\dfrac{1}{36}[\/latex]<\/li>\n
          13. [latex]x=\\dfrac{9-e}{8}[\/latex]<\/li>\n
          14. [latex]n=1[\/latex]<\/li>\n
          15. No solution<\/li>\n
          16. No solution<\/li>\n
          17. [latex]x=\\pm\\dfrac{10}{3}[\/latex]<\/li>\n
          18. [latex]x=10[\/latex]<\/li>\n
          19. [latex]x=0[\/latex]<\/li>\n
          20. [latex]x=\\dfrac{3}{4}[\/latex]<\/li>\n<\/ol>\n

            Exponential and Logarithmic Models<\/h2>\n
              \n
            1. [latex]f(0)\\approx 16.7[\/latex]; The amount initially present is about [latex]16.7[\/latex] units.<\/li>\n
            2. [latex]150[\/latex]<\/li>\n
            3. exponential; [latex]f(x)=1.2^x[\/latex]<\/li>\n
            4. logarithmic
              \n\"Scatter<\/li>\n
            5. logarithmic
              \n\"Scatter<\/li>\n
            6. \"Graph<\/li>\n
            7. about [latex]1.4[\/latex] years<\/li>\n
            8. about [latex]7.3[\/latex] years<\/li>\n
            9. [latex]A=125e^{(-0.3567t)}[\/latex]; [latex]A\\approx 43[\/latex] mg<\/li>\n
            10. about [latex]60[\/latex] days<\/li>\n
            11. [latex]A(t)=250e^{(-0.00822t)}[\/latex]; half-life: about [latex]84[\/latex] minutes<\/li>\n
            12. [latex]r\\approx -0.0667[\/latex], So the hourly decay rate is about [latex]6.67 \\%[\/latex]<\/li>\n
            13. [latex]f(t)=1350e^{(0.03466t)}[\/latex]; after [latex]3[\/latex] hours: [latex]P(180)\\approx 691,200[\/latex]<\/li>\n
            14. [latex]f(t)=256e^{(0.068110t)}[\/latex]; doubling time: about [latex]10[\/latex] minutes<\/li>\n
            15. about [latex]88[\/latex] minutes<\/li>\n
            16. [latex]T(t)=90e^{(-0.008377t)}+75[\/latex], where [latex]t[\/latex] is in minutes.<\/li>\n
            17. about [latex]113[\/latex] minutes<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":106,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/133"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/133\/revisions"}],"predecessor-version":[{"id":141,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/133\/revisions\/141"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/106"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/133\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=133"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=133"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=133"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}