{"id":4864,"date":"2024-10-08T19:35:57","date_gmt":"2024-10-08T19:35:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=4864"},"modified":"2024-11-21T22:07:54","modified_gmt":"2024-11-21T22:07:54","slug":"exponential-and-logarithmic-equations-and-models-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/exponential-and-logarithmic-equations-and-models-get-stronger\/","title":{"raw":"Exponential and Logarithmic Equations and Models: Get Stronger","rendered":"Exponential and Logarithmic Equations and Models: Get Stronger"},"content":{"raw":"

Logarithmic Properties<\/h2>\r\n

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.<\/p>\r\n\r\n

    \r\n \t
  1. [latex]\\log_b(7x \\cdot 2y)[\/latex]<\/li>\r\n \t
  2. [latex]\\log_b\\left(\\dfrac{13}{17}\\right)[\/latex]<\/li>\r\n \t
  3. [latex]\\ln\\left(\\dfrac{1}{4^k}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n

    For the following exercises, condense to a single logarithm if possible.<\/p>\r\n\r\n

      \r\n \t
    1. [latex]\\ln(7) + \\ln(x) + \\ln(y)[\/latex]<\/li>\r\n \t
    2. [latex]\\log_b(28) - \\log_b(7)[\/latex]<\/li>\r\n \t
    3. [latex]-\\log_b\\left(\\dfrac{1}{7}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n

      For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.<\/p>\r\n\r\n

        \r\n \t
      1. [latex]\\log\\left(\\dfrac{x^{15}y^{13}}{z^{19}}\\right)[\/latex]<\/li>\r\n \t
      2. [latex]\\log\\left(\\sqrt{x^3y^{-4}}\\right)[\/latex]<\/li>\r\n \t
      3. [latex]\\log\\left(x^2y^3\\sqrt[3]{x^2y^5}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n

        For the following exercises, condense each expression to a single logarithm using the properties of logarithms.<\/p>\r\n\r\n

          \r\n \t
        1. [latex]\\ln(6x^9) - \\ln(3x^2)[\/latex]<\/li>\r\n \t
        2. [latex]\\log(x) - \\dfrac{1}{2}\\log(y) + 3\\log(z)[\/latex]<\/li>\r\n<\/ol>\r\n

          For the following exercise, rewrite each expression as an equivalent ratio of logs using the indicated base.<\/p>\r\n\r\n

            \r\n \t
          1. [latex]\\log_7(15)[\/latex] to base [latex]e[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, suppose [latex]\\log_5(6) = a[\/latex] and [latex]\\log_5(11) = b[\/latex]. Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of [latex]a[\/latex] and [latex]b[\/latex]. Show the steps for solving.\r\n
              \r\n \t
            1. [latex]\\log_{11}(5)[\/latex]<\/li>\r\n \t
            2. [latex]\\log_{11}\\left(\\dfrac{6}{11}\\right)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use properties of logarithms to evaluate without using a calculator.\r\n
                \r\n \t
              1. [latex]6\\log_8(2) + \\dfrac{\\log_8(64)}{3\\log_8(4)}[\/latex]<\/li>\r\n<\/ol>\r\n

                For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.<\/p>\r\n\r\n

                  \r\n \t
                1. [latex]\\log_3(22)[\/latex]<\/li>\r\n \t
                2. [latex]\\log_6(5.38)[\/latex]<\/li>\r\n \t
                3. [latex]\\log_{\\frac{1}{2}}(4.7)[\/latex]<\/li>\r\n<\/ol>\r\n

                  Exponential and Logarithmic Equations and Models<\/h2>\r\n

                  For the following exercises, use like bases to solve the exponential equation.<\/p>\r\n\r\n

                    \r\n \t
                  1. [latex]64 \\cdot 4^{3x} = 16[\/latex]<\/li>\r\n \t
                  2. [latex]2^{-3n} \\cdot \\dfrac{1}{4} = 2^{n+2}[\/latex]<\/li>\r\n \t
                  3. [latex]\\dfrac{36^{3x}}{36^{2x}} = 216^{2-b}[\/latex]<\/li>\r\n<\/ol>\r\n

                    For the following exercises, use logarithms to solve.<\/p>\r\n\r\n

                      \r\n \t
                    1. [latex]9^{x-10} = 1[\/latex]<\/li>\r\n \t
                    2. [latex]e^{r+10} - 10 = -42[\/latex]<\/li>\r\n \t
                    3. [latex]-8 \\cdot 10^{p+7} - 7 = -24[\/latex]<\/li>\r\n \t
                    4. [latex]e^{-3k} + 6 = 44[\/latex]<\/li>\r\n \t
                    5. [latex]-6e^{9x+8} + 2 = -74[\/latex]<\/li>\r\n \t
                    6. [latex]e^{2x} - e^x - 132 = 0[\/latex]<\/li>\r\n<\/ol>\r\n

                      For the following exercise, use the definition of a logarithm to rewrite the equation as an exponential equation.<\/p>\r\n\r\n

                        \r\n \t
                      1. [latex]\\log\\left(\\dfrac{1}{100}\\right) = -2[\/latex]<\/li>\r\n<\/ol>\r\n

                        For the following exercises, use the definition of a logarithm to solve the equation.<\/p>\r\n\r\n

                          \r\n \t
                        1. [latex]5\\log_7n = 10[\/latex]<\/li>\r\n \t
                        2. [latex]4 + \\log_2(9k) = 2[\/latex]<\/li>\r\n \t
                        3. [latex]10 - 4\\ln(9 - 8x) = 6[\/latex]<\/li>\r\n<\/ol>\r\n

                          For the following exercises, use the one-to-one property of logarithms to solve.<\/p>\r\n\r\n

                            \r\n \t
                          1. [latex]\\log_{13}(5n-2) = \\log_{13}(8-5n)[\/latex]<\/li>\r\n \t
                          2. [latex]\\ln(-3x) = \\ln(x^2-6x)[\/latex]<\/li>\r\n \t
                          3. [latex]\\ln(x-2) - \\ln(x) = \\ln(54)[\/latex]<\/li>\r\n \t
                          4. [latex]\\ln(x^2-10) + \\ln(9) = \\ln(10)[\/latex]<\/li>\r\n<\/ol>\r\n

                            For the following exercises, solve each equation for [latex]x[\/latex].<\/p>\r\n\r\n

                              \r\n \t
                            1. [latex]\\ln(x) + \\ln(x-3) = \\ln(7x)[\/latex]<\/li>\r\n \t
                            2. [latex]\\ln(7) + \\ln(2-4x^2) = \\ln(14)[\/latex]<\/li>\r\n \t
                            3. [latex]\\ln(3) - \\ln(3-3x) = \\ln(4)[\/latex]<\/li>\r\n<\/ol>\r\n

                              Exponential and Logarithmic Models<\/h2>\r\n

                              For the following exercises, use the logistic growth model [latex]f(x)=\\dfrac{150}{1+8e^{-2x}}[\/latex].<\/p>\r\n\r\n

                                \r\n \t
                              1. Find and interpret [latex]f(0)[\/latex]. Round to the nearest tenth.<\/li>\r\n \t
                              2. Find the carrying capacity.<\/li>\r\n \t
                              3. Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                [latex]x[\/latex]<\/th>\r\n[latex]f(x)[\/latex]<\/th>\r\n<\/tr>\r\n
                                [latex]-2[\/latex]<\/td>\r\n[latex]0.694[\/latex]<\/td>\r\n<\/tr>\r\n
                                [latex]-1[\/latex]<\/td>\r\n[latex]0.833[\/latex]<\/td>\r\n<\/tr>\r\n
                                [latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n
                                [latex]1[\/latex]<\/td>\r\n[latex]1.2[\/latex]<\/td>\r\n<\/tr>\r\n
                                [latex]2[\/latex]<\/td>\r\n[latex]1.44[\/latex]<\/td>\r\n<\/tr>\r\n
                                [latex]3[\/latex]<\/td>\r\n[latex]1.728[\/latex]<\/td>\r\n<\/tr>\r\n
                                [latex]4[\/latex]<\/td>\r\n[latex]2.074[\/latex]<\/td>\r\n<\/tr>\r\n
                                [latex]5[\/latex]<\/td>\r\n[latex]2.488[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.\r\n
                                  \r\n \t
                                1. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                  [latex]x[\/latex]<\/th>\r\n[latex]f(x)[\/latex]<\/th>\r\n<\/tr>\r\n
                                  [latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]2[\/latex]<\/td>\r\n[latex]4.079[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]3[\/latex]<\/td>\r\n[latex]5.296[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]4[\/latex]<\/td>\r\n[latex]6.159[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]5[\/latex]<\/td>\r\n[latex]6.828[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]6[\/latex]<\/td>\r\n[latex]7.375[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]7[\/latex]<\/td>\r\n[latex]7.838[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]8[\/latex]<\/td>\r\n[latex]8.238[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]9[\/latex]<\/td>\r\n[latex]8.592[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]10[\/latex]<\/td>\r\n[latex]8.908[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t
                                2. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                  [latex]x[\/latex]<\/th>\r\n[latex]f(x)[\/latex]<\/th>\r\n<\/tr>\r\n
                                  [latex]4[\/latex]<\/td>\r\n[latex]9.429[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]5[\/latex]<\/td>\r\n[latex]9.972[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]6[\/latex]<\/td>\r\n[latex]10.415[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]7[\/latex]<\/td>\r\n[latex]10.79[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]8[\/latex]<\/td>\r\n[latex]11.115[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]9[\/latex]<\/td>\r\n[latex]11.401[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]10[\/latex]<\/td>\r\n[latex]11.657[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]11[\/latex]<\/td>\r\n[latex]11.889[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]12[\/latex]<\/td>\r\n[latex]12.101[\/latex]<\/td>\r\n<\/tr>\r\n
                                  [latex]13[\/latex]<\/td>\r\n[latex]12.295[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n

                                  For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in [latex]t[\/latex] years is modeled by the equation [latex]P(t)=\\dfrac{1000}{1+9e^{-0.6t}}[\/latex].<\/p>\r\n\r\n

                                    \r\n \t
                                  1. Graph the function.<\/li>\r\n \t
                                  2. To the nearest tenth, what is the doubling time for the fish population?<\/li>\r\n \t
                                  3. To the nearest tenth, how long will it take for the population to reach [latex]900[\/latex]?<\/li>\r\n<\/ol>\r\n

                                    For the following exercise, use this scenario: A doctor prescribes [latex]125[\/latex] milligrams of a therapeutic drug that decays by about [latex]30%[\/latex] each hour.<\/p>\r\n\r\n

                                      \r\n \t
                                    1. Write an exponential model representing the amount of the drug remaining in the patient's system after [latex]t[\/latex] hours. Then use the formula to find the amount of the drug that would remain in the patient's system after [latex]3[\/latex] hours. Round to the nearest milligram.<\/li>\r\n<\/ol>\r\n

                                      For the following exercises, use this scenario: A tumor is injected with [latex]0.5[\/latex] grams of Iodine-125, which has a decay rate of [latex]1.15%[\/latex] per day.<\/p>\r\n\r\n

                                        \r\n \t
                                      1. To the nearest day, how long will it take for half of the Iodine-125 to decay?<\/li>\r\n \t
                                      2. A scientist begins with [latex]250[\/latex] grams of a radioactive substance. After [latex]250[\/latex] minutes, the sample has decayed to [latex]32[\/latex] grams. Rounding to five decimal places, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?<\/li>\r\n \t
                                      3. The half-life of Erbium-165 is [latex]10.4[\/latex] hours. What is the hourly decay rate? Express the decimal result to four decimal places and the percentage to two decimal places.<\/li>\r\n \t
                                      4. A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was [latex]1350[\/latex] bacteria. Rounding to five decimal places, write an exponential equation representing this situation. To the nearest whole number, what is the population size after [latex]3[\/latex] hours?<\/li>\r\n<\/ol>\r\n

                                        For the following exercise, use this scenario: A biologist recorded a count of [latex]360[\/latex] bacteria present in a culture after [latex]5[\/latex] minutes and [latex]1000[\/latex] bacteria present after [latex]20[\/latex] minutes.<\/p>\r\n\r\n

                                          \r\n \t
                                        1. Rounding to six decimal places, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?<\/li>\r\n<\/ol>\r\n

                                          For the following exercise, use this scenario: A pot of warm soup with an internal temperature of [latex]100^\\circ[\/latex] Fahrenheit was taken off the stove to cool in a [latex]69^\\circ[\/latex] F room. After fifteen minutes, the internal temperature of the soup was [latex]95^\\circ[\/latex] F.<\/p>\r\n\r\n

                                            \r\n \t
                                          1. To the nearest minute, how long will it take the soup to cool to [latex]80^\\circ[\/latex] F?<\/li>\r\n<\/ol>\r\n

                                            For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of [latex]165^\\circ[\/latex] F and is allowed to cool in a [latex]75^\\circ[\/latex] F room. After half an hour, the internal temperature of the turkey is [latex]145^\\circ[\/latex] F.<\/p>\r\n\r\n

                                              \r\n \t
                                            1. Write a formula that models this situation.<\/li>\r\n \t
                                            2. To the nearest minute, how long will it take the turkey to cool to [latex]110^\\circ[\/latex] F?<\/li>\r\n<\/ol>","rendered":"

                                              Logarithmic Properties<\/h2>\n

                                              For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.<\/p>\n

                                                \n
                                              1. [latex]\\log_b(7x \\cdot 2y)[\/latex]<\/li>\n
                                              2. [latex]\\log_b\\left(\\dfrac{13}{17}\\right)[\/latex]<\/li>\n
                                              3. [latex]\\ln\\left(\\dfrac{1}{4^k}\\right)[\/latex]<\/li>\n<\/ol>\n

                                                For the following exercises, condense to a single logarithm if possible.<\/p>\n

                                                  \n
                                                1. [latex]\\ln(7) + \\ln(x) + \\ln(y)[\/latex]<\/li>\n
                                                2. [latex]\\log_b(28) - \\log_b(7)[\/latex]<\/li>\n
                                                3. [latex]-\\log_b\\left(\\dfrac{1}{7}\\right)[\/latex]<\/li>\n<\/ol>\n

                                                  For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.<\/p>\n

                                                    \n
                                                  1. [latex]\\log\\left(\\dfrac{x^{15}y^{13}}{z^{19}}\\right)[\/latex]<\/li>\n
                                                  2. [latex]\\log\\left(\\sqrt{x^3y^{-4}}\\right)[\/latex]<\/li>\n
                                                  3. [latex]\\log\\left(x^2y^3\\sqrt[3]{x^2y^5}\\right)[\/latex]<\/li>\n<\/ol>\n

                                                    For the following exercises, condense each expression to a single logarithm using the properties of logarithms.<\/p>\n

                                                      \n
                                                    1. [latex]\\ln(6x^9) - \\ln(3x^2)[\/latex]<\/li>\n
                                                    2. [latex]\\log(x) - \\dfrac{1}{2}\\log(y) + 3\\log(z)[\/latex]<\/li>\n<\/ol>\n

                                                      For the following exercise, rewrite each expression as an equivalent ratio of logs using the indicated base.<\/p>\n

                                                        \n
                                                      1. [latex]\\log_7(15)[\/latex] to base [latex]e[\/latex]<\/li>\n<\/ol>\n

                                                        For the following exercises, suppose [latex]\\log_5(6) = a[\/latex] and [latex]\\log_5(11) = b[\/latex]. Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of [latex]a[\/latex] and [latex]b[\/latex]. Show the steps for solving.<\/p>\n

                                                          \n
                                                        1. [latex]\\log_{11}(5)[\/latex]<\/li>\n
                                                        2. [latex]\\log_{11}\\left(\\dfrac{6}{11}\\right)[\/latex]<\/li>\n<\/ol>\n

                                                          For the following exercises, use properties of logarithms to evaluate without using a calculator.<\/p>\n

                                                            \n
                                                          1. [latex]6\\log_8(2) + \\dfrac{\\log_8(64)}{3\\log_8(4)}[\/latex]<\/li>\n<\/ol>\n

                                                            For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.<\/p>\n

                                                              \n
                                                            1. [latex]\\log_3(22)[\/latex]<\/li>\n
                                                            2. [latex]\\log_6(5.38)[\/latex]<\/li>\n
                                                            3. [latex]\\log_{\\frac{1}{2}}(4.7)[\/latex]<\/li>\n<\/ol>\n

                                                              Exponential and Logarithmic Equations and Models<\/h2>\n

                                                              For the following exercises, use like bases to solve the exponential equation.<\/p>\n

                                                                \n
                                                              1. [latex]64 \\cdot 4^{3x} = 16[\/latex]<\/li>\n
                                                              2. [latex]2^{-3n} \\cdot \\dfrac{1}{4} = 2^{n+2}[\/latex]<\/li>\n
                                                              3. [latex]\\dfrac{36^{3x}}{36^{2x}} = 216^{2-b}[\/latex]<\/li>\n<\/ol>\n

                                                                For the following exercises, use logarithms to solve.<\/p>\n

                                                                  \n
                                                                1. [latex]9^{x-10} = 1[\/latex]<\/li>\n
                                                                2. [latex]e^{r+10} - 10 = -42[\/latex]<\/li>\n
                                                                3. [latex]-8 \\cdot 10^{p+7} - 7 = -24[\/latex]<\/li>\n
                                                                4. [latex]e^{-3k} + 6 = 44[\/latex]<\/li>\n
                                                                5. [latex]-6e^{9x+8} + 2 = -74[\/latex]<\/li>\n
                                                                6. [latex]e^{2x} - e^x - 132 = 0[\/latex]<\/li>\n<\/ol>\n

                                                                  For the following exercise, use the definition of a logarithm to rewrite the equation as an exponential equation.<\/p>\n

                                                                    \n
                                                                  1. [latex]\\log\\left(\\dfrac{1}{100}\\right) = -2[\/latex]<\/li>\n<\/ol>\n

                                                                    For the following exercises, use the definition of a logarithm to solve the equation.<\/p>\n

                                                                      \n
                                                                    1. [latex]5\\log_7n = 10[\/latex]<\/li>\n
                                                                    2. [latex]4 + \\log_2(9k) = 2[\/latex]<\/li>\n
                                                                    3. [latex]10 - 4\\ln(9 - 8x) = 6[\/latex]<\/li>\n<\/ol>\n

                                                                      For the following exercises, use the one-to-one property of logarithms to solve.<\/p>\n

                                                                        \n
                                                                      1. [latex]\\log_{13}(5n-2) = \\log_{13}(8-5n)[\/latex]<\/li>\n
                                                                      2. [latex]\\ln(-3x) = \\ln(x^2-6x)[\/latex]<\/li>\n
                                                                      3. [latex]\\ln(x-2) - \\ln(x) = \\ln(54)[\/latex]<\/li>\n
                                                                      4. [latex]\\ln(x^2-10) + \\ln(9) = \\ln(10)[\/latex]<\/li>\n<\/ol>\n

                                                                        For the following exercises, solve each equation for [latex]x[\/latex].<\/p>\n

                                                                          \n
                                                                        1. [latex]\\ln(x) + \\ln(x-3) = \\ln(7x)[\/latex]<\/li>\n
                                                                        2. [latex]\\ln(7) + \\ln(2-4x^2) = \\ln(14)[\/latex]<\/li>\n
                                                                        3. [latex]\\ln(3) - \\ln(3-3x) = \\ln(4)[\/latex]<\/li>\n<\/ol>\n

                                                                          Exponential and Logarithmic Models<\/h2>\n

                                                                          For the following exercises, use the logistic growth model [latex]f(x)=\\dfrac{150}{1+8e^{-2x}}[\/latex].<\/p>\n

                                                                            \n
                                                                          1. Find and interpret [latex]f(0)[\/latex]. Round to the nearest tenth.<\/li>\n
                                                                          2. Find the carrying capacity.<\/li>\n
                                                                          3. Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data.
                                                                            \n\n\n\n\n\n\n\n\n\n\n\n
                                                                            [latex]x[\/latex]<\/th>\n[latex]f(x)[\/latex]<\/th>\n<\/tr>\n
                                                                            [latex]-2[\/latex]<\/td>\n[latex]0.694[\/latex]<\/td>\n<\/tr>\n
                                                                            [latex]-1[\/latex]<\/td>\n[latex]0.833[\/latex]<\/td>\n<\/tr>\n
                                                                            [latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n<\/tr>\n
                                                                            [latex]1[\/latex]<\/td>\n[latex]1.2[\/latex]<\/td>\n<\/tr>\n
                                                                            [latex]2[\/latex]<\/td>\n[latex]1.44[\/latex]<\/td>\n<\/tr>\n
                                                                            [latex]3[\/latex]<\/td>\n[latex]1.728[\/latex]<\/td>\n<\/tr>\n
                                                                            [latex]4[\/latex]<\/td>\n[latex]2.074[\/latex]<\/td>\n<\/tr>\n
                                                                            [latex]5[\/latex]<\/td>\n[latex]2.488[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                                                                            For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.<\/p>\n

                                                                              \n
                                                                            1. \n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                                                              [latex]x[\/latex]<\/th>\n[latex]f(x)[\/latex]<\/th>\n<\/tr>\n
                                                                              [latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]2[\/latex]<\/td>\n[latex]4.079[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]3[\/latex]<\/td>\n[latex]5.296[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]4[\/latex]<\/td>\n[latex]6.159[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]5[\/latex]<\/td>\n[latex]6.828[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]6[\/latex]<\/td>\n[latex]7.375[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]7[\/latex]<\/td>\n[latex]7.838[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]8[\/latex]<\/td>\n[latex]8.238[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]9[\/latex]<\/td>\n[latex]8.592[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]10[\/latex]<\/td>\n[latex]8.908[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n
                                                                            2. \n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                                                              [latex]x[\/latex]<\/th>\n[latex]f(x)[\/latex]<\/th>\n<\/tr>\n
                                                                              [latex]4[\/latex]<\/td>\n[latex]9.429[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]5[\/latex]<\/td>\n[latex]9.972[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]6[\/latex]<\/td>\n[latex]10.415[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]7[\/latex]<\/td>\n[latex]10.79[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]8[\/latex]<\/td>\n[latex]11.115[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]9[\/latex]<\/td>\n[latex]11.401[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]10[\/latex]<\/td>\n[latex]11.657[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]11[\/latex]<\/td>\n[latex]11.889[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]12[\/latex]<\/td>\n[latex]12.101[\/latex]<\/td>\n<\/tr>\n
                                                                              [latex]13[\/latex]<\/td>\n[latex]12.295[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                                                                              For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in [latex]t[\/latex] years is modeled by the equation [latex]P(t)=\\dfrac{1000}{1+9e^{-0.6t}}[\/latex].<\/p>\n

                                                                                \n
                                                                              1. Graph the function.<\/li>\n
                                                                              2. To the nearest tenth, what is the doubling time for the fish population?<\/li>\n
                                                                              3. To the nearest tenth, how long will it take for the population to reach [latex]900[\/latex]?<\/li>\n<\/ol>\n

                                                                                For the following exercise, use this scenario: A doctor prescribes [latex]125[\/latex] milligrams of a therapeutic drug that decays by about [latex]30%[\/latex] each hour.<\/p>\n

                                                                                  \n
                                                                                1. Write an exponential model representing the amount of the drug remaining in the patient’s system after [latex]t[\/latex] hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after [latex]3[\/latex] hours. Round to the nearest milligram.<\/li>\n<\/ol>\n

                                                                                  For the following exercises, use this scenario: A tumor is injected with [latex]0.5[\/latex] grams of Iodine-125, which has a decay rate of [latex]1.15%[\/latex] per day.<\/p>\n

                                                                                    \n
                                                                                  1. To the nearest day, how long will it take for half of the Iodine-125 to decay?<\/li>\n
                                                                                  2. A scientist begins with [latex]250[\/latex] grams of a radioactive substance. After [latex]250[\/latex] minutes, the sample has decayed to [latex]32[\/latex] grams. Rounding to five decimal places, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?<\/li>\n
                                                                                  3. The half-life of Erbium-165 is [latex]10.4[\/latex] hours. What is the hourly decay rate? Express the decimal result to four decimal places and the percentage to two decimal places.<\/li>\n
                                                                                  4. A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was [latex]1350[\/latex] bacteria. Rounding to five decimal places, write an exponential equation representing this situation. To the nearest whole number, what is the population size after [latex]3[\/latex] hours?<\/li>\n<\/ol>\n

                                                                                    For the following exercise, use this scenario: A biologist recorded a count of [latex]360[\/latex] bacteria present in a culture after [latex]5[\/latex] minutes and [latex]1000[\/latex] bacteria present after [latex]20[\/latex] minutes.<\/p>\n

                                                                                      \n
                                                                                    1. Rounding to six decimal places, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?<\/li>\n<\/ol>\n

                                                                                      For the following exercise, use this scenario: A pot of warm soup with an internal temperature of [latex]100^\\circ[\/latex] Fahrenheit was taken off the stove to cool in a [latex]69^\\circ[\/latex] F room. After fifteen minutes, the internal temperature of the soup was [latex]95^\\circ[\/latex] F.<\/p>\n

                                                                                        \n
                                                                                      1. To the nearest minute, how long will it take the soup to cool to [latex]80^\\circ[\/latex] F?<\/li>\n<\/ol>\n

                                                                                        For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of [latex]165^\\circ[\/latex] F and is allowed to cool in a [latex]75^\\circ[\/latex] F room. After half an hour, the internal temperature of the turkey is [latex]145^\\circ[\/latex] F.<\/p>\n

                                                                                          \n
                                                                                        1. Write a formula that models this situation.<\/li>\n
                                                                                        2. To the nearest minute, how long will it take the turkey to cool to [latex]110^\\circ[\/latex] F?<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":24,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":280,"module-header":"practice","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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4864"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4864\/revisions"}],"predecessor-version":[{"id":6007,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4864\/revisions\/6007"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/280"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4864\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=4864"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4864"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=4864"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=4864"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}