{"id":692,"date":"2025-07-14T21:07:48","date_gmt":"2025-07-14T21:07:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=692"},"modified":"2026-01-15T16:38:28","modified_gmt":"2026-01-15T16:38:28","slug":"module-8-exponential-and-logarithmic-equations-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-8-exponential-and-logarithmic-equations-cheat-sheet\/","title":{"raw":"Exponential and Logarithmic Equations: Cheat Sheet","rendered":"Exponential and Logarithmic Equations: Cheat Sheet"},"content":{"raw":"

Essential Concepts<\/h2>\r\n

Logarithmic Properties<\/h3>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Property<\/th>\r\nFormula<\/th>\r\nMeaning<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Zero Property<\/strong><\/td>\r\n[latex]\\log_b(1) = 0[\/latex]<\/td>\r\nThe logarithm of 1 to any base is 0 (because [latex]b^0 = 1[\/latex])<\/td>\r\n<\/tr>\r\n
Identity Property<\/strong><\/td>\r\n[latex]\\log_b(b) = 1[\/latex]<\/td>\r\nThe logarithm of the base to itself is 1 (because [latex]b^1 = b[\/latex])<\/td>\r\n<\/tr>\r\n
Inverse Property<\/strong><\/td>\r\n[latex]b^{\\log_b(x)} = x[\/latex] and [latex]\\log_b(b^x) = x[\/latex]<\/td>\r\nLogarithms and exponentials undo each other<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Rule<\/th>\r\nFormula<\/th>\r\nDescription<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Product Rule<\/strong><\/td>\r\n[latex]\\log_b(MN) = \\log_b(M) + \\log_b(N)[\/latex]<\/td>\r\nThe log of a product equals the sum of the logs<\/td>\r\n<\/tr>\r\n
Quotient Rule<\/strong><\/td>\r\n[latex]\\log_b\\left(\\frac{M}{N}\\right) = \\log_b(M) - \\log_b(N)[\/latex]<\/td>\r\nThe log of a quotient equals the difference of the logs<\/td>\r\n<\/tr>\r\n
Power Rule<\/strong><\/td>\r\n[latex]\\log_b(M^n) = n\\log_b(M)[\/latex]<\/td>\r\nThe log of a power equals the exponent times the log<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

Note: These rules only apply to products, quotients, powers, and roots\u2014never to addition or subtraction inside the argument<\/em><\/p>\r\n

Change-of-Base Formula<\/strong><\/p>\r\n

For any positive real numbers [latex]M[\/latex], [latex]b[\/latex], and [latex]n[\/latex] (where [latex]n \\neq 1[\/latex] and [latex]b \\neq 1[\/latex]):<\/p>\r\n\r\n

    \r\n \t
  • [latex]\\log_b(M) = \\frac{\\ln(M)}{\\ln(b)}[\/latex] (using natural log)<\/li>\r\n \t
  • [latex]\\log_b(M) = \\frac{\\log(M)}{\\log(b)}[\/latex] (using common log)<\/li>\r\n<\/ul>\r\n

    Exponential and Logarithmic Equations<\/h3>\r\n

    Solving Exponential Equations Using Like Bases<\/strong><\/p>\r\n

    One-to-One Property of Exponential Functions:<\/strong> For any algebraic expressions [latex]S[\/latex] and [latex]T[\/latex], and any positive real number [latex]b \\neq 1[\/latex]:<\/p>\r\n

    [latex]b^S = b^T[\/latex] if and only if [latex]S = T[\/latex]<\/p>\r\n

    Strategy:<\/strong><\/p>\r\n\r\n

      \r\n \t
    1. Rewrite each side with a common base using exponent properties<\/li>\r\n \t
    2. Apply the one-to-one property to set exponents equal<\/li>\r\n \t
    3. Solve the resulting equation<\/li>\r\n<\/ol>\r\n

      Solving Exponential Equations Using Logarithms<\/strong><\/p>\r\n

      Property of Logarithmic Equality:<\/strong> If [latex]\\log_b(M) = \\log_b(N)[\/latex], then [latex]M = N[\/latex]<\/p>\r\n

      When bases cannot be made equal:<\/p>\r\n\r\n

        \r\n \t
      1. Apply logarithm to both sides (use [latex]\\ln[\/latex] if base [latex]e[\/latex] is present, [latex]\\log[\/latex] if base 10 is present, or either otherwise)<\/li>\r\n \t
      2. Use the power rule to bring exponents down<\/li>\r\n \t
      3. Solve for the unknown<\/li>\r\n<\/ol>\r\n

        Equations with Base e<\/strong><\/p>\r\n

        For [latex]Ae^{kt} = c[\/latex]:<\/p>\r\n\r\n

          \r\n \t
        1. Isolate exponential: [latex]e^{kt} = \\frac{c}{A}[\/latex]<\/li>\r\n \t
        2. Apply [latex]\\ln[\/latex]: [latex]kt = \\ln\\left(\\frac{c}{A}\\right)[\/latex]<\/li>\r\n \t
        3. Solve for variable<\/li>\r\n<\/ol>\r\n

          Remember:<\/em> [latex]\\ln(e^x) = x[\/latex] and [latex]e^{\\ln(x)} = x[\/latex] for [latex]x > 0[\/latex]<\/p>\r\n

          Solving Logarithmic Equations Using the Definition<\/strong><\/p>\r\n

          For [latex]\\log_b(S) = c[\/latex], convert to exponential form: [latex]b^c = S[\/latex]<\/p>\r\n

          Strategy:<\/strong><\/p>\r\n\r\n

            \r\n \t
          1. Use properties to write as a single log on one side<\/li>\r\n \t
          2. Convert to exponential form<\/li>\r\n \t
          3. Solve the equation<\/li>\r\n<\/ol>\r\n

            Solving Logarithmic Equations Using the One-to-One Property<\/strong><\/p>\r\n

            One-to-One Property of Logarithms:<\/strong> For [latex]x > 0[\/latex], [latex]S > 0[\/latex], [latex]T > 0[\/latex], and [latex]b > 0[\/latex] (where [latex]b \\neq 1[\/latex]):<\/p>\r\n

            [latex]\\log_b(S) = \\log_b(T)[\/latex] if and only if [latex]S = T[\/latex]<\/p>\r\n

            Strategy:<\/strong><\/p>\r\n\r\n

              \r\n \t
            1. Use properties to get single logs with same base on each side<\/li>\r\n \t
            2. Set arguments equal<\/li>\r\n \t
            3. Solve the equation<\/li>\r\n \t
            4. Always check for extraneous solutions<\/strong><\/li>\r\n<\/ol>\r\n

              Extraneous Solutions<\/strong><\/p>\r\n

              Solutions are extraneous when:<\/p>\r\n\r\n

                \r\n \t
              • The logarithm of a negative number or zero would be required<\/li>\r\n \t
              • The solution doesn't satisfy the original equation<\/li>\r\n<\/ul>\r\n

                Exponential and Logarithmic Models<\/h3>\r\n

                Continuous Growth\/Decay Model:<\/strong> [latex]A(t) = A_0 e^{kt}[\/latex]<\/p>\r\n

                Where:<\/p>\r\n\r\n

                  \r\n \t
                • [latex]A_0[\/latex] = initial amount at [latex]t = 0[\/latex]<\/li>\r\n \t
                • [latex]k[\/latex] = continuous growth rate ([latex]k > 0[\/latex] for growth, [latex]k < 0[\/latex] for decay)<\/li>\r\n \t
                • [latex]t[\/latex] = time<\/li>\r\n \t
                • [latex]A(t)[\/latex] = amount at time [latex]t[\/latex]<\/li>\r\n<\/ul>\r\n

                  Characteristics:<\/strong><\/p>\r\n\r\n

                    \r\n \t
                  • Domain: [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t
                  • Range: [latex](0, \\infty)[\/latex]<\/li>\r\n \t
                  • Horizontal asymptote: [latex]y = 0[\/latex]<\/li>\r\n<\/ul>\r\n

                    Half-Life<\/strong><\/p>\r\n

                    Time for a quantity to decay to half its original amount:<\/p>\r\n

                    [latex]t = \\frac{\\ln(0.5)}{k} = -\\frac{\\ln(2)}{k}[\/latex]<\/p>\r\n

                    Alternative formula:<\/strong> [latex]A(t) = A_0\\left(\\frac{1}{2}\\right)^{\\frac{t}{T}}[\/latex] where [latex]T[\/latex] is the half-life<\/p>\r\n

                    Use:<\/em> Common in radioactive decay and carbon-14 dating<\/p>\r\n

                    Newton's Law of Cooling<\/strong><\/p>\r\n

                    Temperature of an object in surrounding air:<\/p>\r\n

                    [latex]T(t) = Ae^{kt} + T_s[\/latex]<\/p>\r\n

                    Where:<\/p>\r\n\r\n

                      \r\n \t
                    • [latex]A[\/latex] = difference between initial and surrounding temperature<\/li>\r\n \t
                    • [latex]k[\/latex] = continuous rate of cooling (negative)<\/li>\r\n \t
                    • [latex]T_s[\/latex] = surrounding (ambient) temperature<\/li>\r\n \t
                    • [latex]t[\/latex] = time<\/li>\r\n<\/ul>\r\n

                      Strategy:<\/strong><\/p>\r\n\r\n

                        \r\n \t
                      1. Set [latex]T_s[\/latex] equal to ambient temperature<\/li>\r\n \t
                      2. Use initial conditions to find [latex]A[\/latex]<\/li>\r\n \t
                      3. Use a second data point to find [latex]k[\/latex]<\/li>\r\n \t
                      4. Use model to make predictions<\/li>\r\n<\/ol>\r\n

                        Logistic Growth Model:<\/strong> [latex]f(x) = \\frac{c}{1 + ae^{-bx}}[\/latex]<\/p>\r\n

                        Where:<\/p>\r\n\r\n

                          \r\n \t
                        • [latex]\\frac{c}{1+a}[\/latex] = initial value<\/li>\r\n \t
                        • [latex]c[\/latex] = carrying capacity (limiting value\/maximum)<\/li>\r\n \t
                        • [latex]b[\/latex] = constant determined by rate of growth<\/li>\r\n<\/ul>\r\n

                          Choosing the Right Model<\/strong><\/p>\r\n\r\n

                          \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                          Model<\/th>\r\nWhen to Use<\/th>\r\nConcavity<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
                          Exponential<\/strong><\/td>\r\nRapid growth\/decay without bound<\/td>\r\nAlways concave up<\/td>\r\n<\/tr>\r\n
                          Logarithmic<\/strong><\/td>\r\nRapid change at first, then slows<\/td>\r\nAlways concave down<\/td>\r\n<\/tr>\r\n
                          Logistic<\/strong><\/td>\r\nGrowth with upper limit<\/td>\r\nChanges from concave up to concave down<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

                          Fitting Exponential Models to Data<\/h3>\r\n

                          Using a Graphing Calculator for Regression:<\/strong><\/p>\r\n\r\n

                            \r\n \t
                          1. Enter data in lists (L1 for input, L2 for output)<\/li>\r\n \t
                          2. Create scatter plot to identify pattern<\/li>\r\n \t
                          3. Use appropriate regression command:\r\n
                              \r\n \t
                            • ExpReg<\/strong> for exponential<\/li>\r\n \t
                            • LnReg<\/strong> for logarithmic<\/li>\r\n \t
                            • Logistic<\/strong> for logistic<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
                            • Graph model with data to verify fit<\/li>\r\n \t
                            • Check [latex]r^2[\/latex] value (closer to 1 = better fit)<\/li>\r\n<\/ol>\r\n

                              Interpolation vs. Extrapolation<\/strong><\/p>\r\n\r\n

                                \r\n \t
                              • Interpolation:<\/strong> Predictions within the data range (more reliable)<\/li>\r\n \t
                              • Extrapolation:<\/strong> Predictions outside the data range (less reliable, requires careful reasoning)<\/li>\r\n<\/ul>\r\n

                                Key Equations<\/h2>\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\r\n\r\n
                                Product rule for logarithms<\/strong><\/td>\r\n[latex]\\log_b(MN) = \\log_b(M) + \\log_b(N)[\/latex]<\/td>\r\n<\/tr>\r\n
                                Quotient rule for logarithms<\/strong><\/td>\r\n[latex]\\log_b\\left(\\frac{M}{N}\\right) = \\log_b(M) - \\log_b(N)[\/latex]<\/td>\r\n<\/tr>\r\n
                                Power rule for logarithms<\/strong><\/td>\r\n[latex]\\log_b(M^n) = n\\log_b(M)[\/latex]<\/td>\r\n<\/tr>\r\n
                                Change-of-base formula<\/strong><\/td>\r\n[latex]\\log_b(M) = \\frac{\\log_n(M)}{\\log_n(b)} = \\frac{\\ln(M)}{\\ln(b)}[\/latex]<\/td>\r\n<\/tr>\r\n
                                One-to-one property (exponential)<\/strong><\/td>\r\n[latex]b^S = b^T \\Leftrightarrow S = T[\/latex]<\/td>\r\n<\/tr>\r\n
                                One-to-one property (logarithmic)<\/strong><\/td>\r\n[latex]\\log_b(S) = \\log_b(T) \\Leftrightarrow S = T[\/latex]<\/td>\r\n<\/tr>\r\n
                                Continuous growth\/decay<\/strong><\/td>\r\n[latex]A(t) = A_0 e^{kt}[\/latex]<\/td>\r\n<\/tr>\r\n
                                Doubling time<\/strong><\/td>\r\n[latex]t = \\frac{\\ln(2)}{k}[\/latex]<\/td>\r\n<\/tr>\r\n
                                Half-life<\/strong><\/td>\r\n[latex]t = -\\frac{\\ln(2)}{k}[\/latex] or [latex]A(t) = A_0\\left(\\frac{1}{2}\\right)^{\\frac{t}{T}}[\/latex]<\/td>\r\n<\/tr>\r\n
                                Newton's Law of Cooling<\/strong><\/td>\r\n[latex]T(t) = Ae^{kt} + T_s[\/latex]<\/td>\r\n<\/tr>\r\n
                                Logistic growth<\/strong><\/td>\r\n[latex]f(x) = \\frac{c}{1 + ae^{-bx}}[\/latex]<\/td>\r\n<\/tr>\r\n
                                Exponential regression<\/strong><\/td>\r\n[latex]y = ab^x[\/latex]<\/td>\r\n<\/tr>\r\n
                                Logarithmic regression<\/strong><\/td>\r\n[latex]y = a + b\\ln(x)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                                Glossary<\/h2>\r\n

                                carrying capacity<\/strong><\/p>\r\n

                                The limiting value [latex]c[\/latex] in a logistic model; represents the maximum sustainable population or quantity.<\/p>\r\n

                                change-of-base formula<\/strong><\/p>\r\n

                                A formula that allows evaluation of logarithms with any base using logarithms of another base: [latex]\\log_b(M) = \\frac{\\log_n(M)}{\\log_n(b)}[\/latex].<\/p>\r\n

                                concavity<\/strong><\/p>\r\n

                                The direction a curve bends; concave up curves bend upward (can hold water), concave down curves bend downward (cannot hold water).<\/p>\r\n

                                continuous growth\/decay model<\/strong><\/p>\r\n

                                A model of the form [latex]A(t) = A_0 e^{kt}[\/latex] where [latex]k > 0[\/latex] represents growth and [latex]k < 0[\/latex] represents decay.<\/p>\r\n

                                extraneous solution<\/strong><\/p>\r\n

                                A solution that emerges algebraically but doesn't satisfy the original equation; common when logarithms of negative numbers or zero would be required.<\/p>\r\n

                                extrapolation<\/strong><\/p>\r\n

                                Using a model to make predictions outside the range of original data; less reliable and requires careful reasoning.<\/p>\r\n

                                half-life<\/strong><\/p>\r\n

                                The time required for an exponentially decaying quantity to reduce to half its original amount; calculated as [latex]t = -\\frac{\\ln(2)}{k}[\/latex].<\/p>\r\n

                                identity property of logarithms<\/strong><\/p>\r\n

                                The logarithm of the base to itself equals 1: [latex]\\log_b(b) = 1[\/latex].<\/p>\r\n

                                interpolation<\/strong><\/p>\r\n

                                Using a model to make predictions within the range of original data; generally more reliable than extrapolation.<\/p>\r\n

                                inverse property of logarithms<\/strong><\/p>\r\n

                                Logarithms and exponentials undo each other: [latex]b^{\\log_b(x)} = x[\/latex] and [latex]\\log_b(b^x) = x[\/latex].<\/p>\r\n

                                logistic growth<\/strong><\/p>\r\n

                                Growth that is exponential at first but slows as it approaches a maximum value (carrying capacity); modeled by [latex]f(x) = \\frac{c}{1 + ae^{-bx}}[\/latex].<\/p>\r\n

                                Newton's Law of Cooling<\/strong><\/p>\r\n

                                A model describing how an object's temperature changes over time to equalize with surrounding temperature: [latex]T(t) = Ae^{kt} + T_s[\/latex].<\/p>\r\n

                                one-to-one property of exponential functions<\/strong><\/p>\r\n

                                If [latex]b^S = b^T[\/latex], then [latex]S = T[\/latex] for any positive real number [latex]b \\neq 1[\/latex].<\/p>\r\n

                                one-to-one property of logarithmic functions<\/strong><\/p>\r\n

                                If [latex]\\log_b(S) = \\log_b(T)[\/latex], then [latex]S = T[\/latex] for positive real numbers [latex]S[\/latex], [latex]T[\/latex], and base [latex]b > 0[\/latex], [latex]b \\neq 1[\/latex].<\/p>\r\n

                                power rule for logarithms<\/strong><\/p>\r\n

                                The logarithm of a power equals the exponent times the logarithm: [latex]\\log_b(M^n) = n\\log_b(M)[\/latex].<\/p>\r\n

                                product rule for logarithms<\/strong><\/p>\r\n

                                The logarithm of a product equals the sum of the logarithms: [latex]\\log_b(MN) = \\log_b(M) + \\log_b(N)[\/latex].<\/p>\r\n

                                property of logarithmic equality<\/strong><\/p>\r\n

                                If [latex]\\log_b(M) = \\log_b(N)[\/latex], then [latex]M = N[\/latex] for any positive real numbers [latex]M[\/latex], [latex]N[\/latex], and base [latex]b > 0[\/latex], [latex]b \\neq 1[\/latex].<\/p>\r\n

                                quotient rule for logarithms<\/strong><\/p>\r\n

                                The logarithm of a quotient equals the difference of the logarithms: [latex]\\log_b\\left(\\frac{M}{N}\\right) = \\log_b(M) - \\log_b(N)[\/latex].<\/p>\r\n

                                radiocarbon dating<\/strong><\/p>\r\n

                                A method for determining the age of organic materials by measuring the ratio of carbon-14 to carbon-12, based on carbon-14's half-life of 5,730 years.<\/p>\r\n

                                regression<\/strong><\/p>\r\n

                                A method of fitting an algebraic model to data<\/p>\r\n

                                zero property of logarithms<\/strong><\/p>\r\n

                                The logarithm of 1 to any base equals 0: [latex]\\log_b(1) = 0[\/latex].<\/p>","rendered":"

                                Essential Concepts<\/h2>\n

                                Logarithmic Properties<\/h3>\n
                                \n\n\n\n\n\n\n\n
                                Property<\/th>\nFormula<\/th>\nMeaning<\/th>\n<\/tr>\n<\/thead>\n
                                Zero Property<\/strong><\/td>\n[latex]\\log_b(1) = 0[\/latex]<\/td>\nThe logarithm of 1 to any base is 0 (because [latex]b^0 = 1[\/latex])<\/td>\n<\/tr>\n
                                Identity Property<\/strong><\/td>\n[latex]\\log_b(b) = 1[\/latex]<\/td>\nThe logarithm of the base to itself is 1 (because [latex]b^1 = b[\/latex])<\/td>\n<\/tr>\n
                                Inverse Property<\/strong><\/td>\n[latex]b^{\\log_b(x)} = x[\/latex] and [latex]\\log_b(b^x) = x[\/latex]<\/td>\nLogarithms and exponentials undo each other<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
                                \n\n\n\n\n\n\n\n
                                Rule<\/th>\nFormula<\/th>\nDescription<\/th>\n<\/tr>\n<\/thead>\n
                                Product Rule<\/strong><\/td>\n[latex]\\log_b(MN) = \\log_b(M) + \\log_b(N)[\/latex]<\/td>\nThe log of a product equals the sum of the logs<\/td>\n<\/tr>\n
                                Quotient Rule<\/strong><\/td>\n[latex]\\log_b\\left(\\frac{M}{N}\\right) = \\log_b(M) - \\log_b(N)[\/latex]<\/td>\nThe log of a quotient equals the difference of the logs<\/td>\n<\/tr>\n
                                Power Rule<\/strong><\/td>\n[latex]\\log_b(M^n) = n\\log_b(M)[\/latex]<\/td>\nThe log of a power equals the exponent times the log<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                Note: These rules only apply to products, quotients, powers, and roots\u2014never to addition or subtraction inside the argument<\/em><\/p>\n

                                Change-of-Base Formula<\/strong><\/p>\n

                                For any positive real numbers [latex]M[\/latex], [latex]b[\/latex], and [latex]n[\/latex] (where [latex]n \\neq 1[\/latex] and [latex]b \\neq 1[\/latex]):<\/p>\n

                                  \n
                                • [latex]\\log_b(M) = \\frac{\\ln(M)}{\\ln(b)}[\/latex] (using natural log)<\/li>\n
                                • [latex]\\log_b(M) = \\frac{\\log(M)}{\\log(b)}[\/latex] (using common log)<\/li>\n<\/ul>\n

                                  Exponential and Logarithmic Equations<\/h3>\n

                                  Solving Exponential Equations Using Like Bases<\/strong><\/p>\n

                                  One-to-One Property of Exponential Functions:<\/strong> For any algebraic expressions [latex]S[\/latex] and [latex]T[\/latex], and any positive real number [latex]b \\neq 1[\/latex]:<\/p>\n

                                  [latex]b^S = b^T[\/latex] if and only if [latex]S = T[\/latex]<\/p>\n

                                  Strategy:<\/strong><\/p>\n

                                    \n
                                  1. Rewrite each side with a common base using exponent properties<\/li>\n
                                  2. Apply the one-to-one property to set exponents equal<\/li>\n
                                  3. Solve the resulting equation<\/li>\n<\/ol>\n

                                    Solving Exponential Equations Using Logarithms<\/strong><\/p>\n

                                    Property of Logarithmic Equality:<\/strong> If [latex]\\log_b(M) = \\log_b(N)[\/latex], then [latex]M = N[\/latex]<\/p>\n

                                    When bases cannot be made equal:<\/p>\n

                                      \n
                                    1. Apply logarithm to both sides (use [latex]\\ln[\/latex] if base [latex]e[\/latex] is present, [latex]\\log[\/latex] if base 10 is present, or either otherwise)<\/li>\n
                                    2. Use the power rule to bring exponents down<\/li>\n
                                    3. Solve for the unknown<\/li>\n<\/ol>\n

                                      Equations with Base e<\/strong><\/p>\n

                                      For [latex]Ae^{kt} = c[\/latex]:<\/p>\n

                                        \n
                                      1. Isolate exponential: [latex]e^{kt} = \\frac{c}{A}[\/latex]<\/li>\n
                                      2. Apply [latex]\\ln[\/latex]: [latex]kt = \\ln\\left(\\frac{c}{A}\\right)[\/latex]<\/li>\n
                                      3. Solve for variable<\/li>\n<\/ol>\n

                                        Remember:<\/em> [latex]\\ln(e^x) = x[\/latex] and [latex]e^{\\ln(x)} = x[\/latex] for [latex]x > 0[\/latex]<\/p>\n

                                        Solving Logarithmic Equations Using the Definition<\/strong><\/p>\n

                                        For [latex]\\log_b(S) = c[\/latex], convert to exponential form: [latex]b^c = S[\/latex]<\/p>\n

                                        Strategy:<\/strong><\/p>\n

                                          \n
                                        1. Use properties to write as a single log on one side<\/li>\n
                                        2. Convert to exponential form<\/li>\n
                                        3. Solve the equation<\/li>\n<\/ol>\n

                                          Solving Logarithmic Equations Using the One-to-One Property<\/strong><\/p>\n

                                          One-to-One Property of Logarithms:<\/strong> For [latex]x > 0[\/latex], [latex]S > 0[\/latex], [latex]T > 0[\/latex], and [latex]b > 0[\/latex] (where [latex]b \\neq 1[\/latex]):<\/p>\n

                                          [latex]\\log_b(S) = \\log_b(T)[\/latex] if and only if [latex]S = T[\/latex]<\/p>\n

                                          Strategy:<\/strong><\/p>\n

                                            \n
                                          1. Use properties to get single logs with same base on each side<\/li>\n
                                          2. Set arguments equal<\/li>\n
                                          3. Solve the equation<\/li>\n
                                          4. Always check for extraneous solutions<\/strong><\/li>\n<\/ol>\n

                                            Extraneous Solutions<\/strong><\/p>\n

                                            Solutions are extraneous when:<\/p>\n

                                              \n
                                            • The logarithm of a negative number or zero would be required<\/li>\n
                                            • The solution doesn’t satisfy the original equation<\/li>\n<\/ul>\n

                                              Exponential and Logarithmic Models<\/h3>\n

                                              Continuous Growth\/Decay Model:<\/strong> [latex]A(t) = A_0 e^{kt}[\/latex]<\/p>\n

                                              Where:<\/p>\n

                                                \n
                                              • [latex]A_0[\/latex] = initial amount at [latex]t = 0[\/latex]<\/li>\n
                                              • [latex]k[\/latex] = continuous growth rate ([latex]k > 0[\/latex] for growth, [latex]k < 0[\/latex] for decay)<\/li>\n
                                              • [latex]t[\/latex] = time<\/li>\n
                                              • [latex]A(t)[\/latex] = amount at time [latex]t[\/latex]<\/li>\n<\/ul>\n

                                                Characteristics:<\/strong><\/p>\n

                                                  \n
                                                • Domain: [latex](-\\infty, \\infty)[\/latex]<\/li>\n
                                                • Range: [latex](0, \\infty)[\/latex]<\/li>\n
                                                • Horizontal asymptote: [latex]y = 0[\/latex]<\/li>\n<\/ul>\n

                                                  Half-Life<\/strong><\/p>\n

                                                  Time for a quantity to decay to half its original amount:<\/p>\n

                                                  [latex]t = \\frac{\\ln(0.5)}{k} = -\\frac{\\ln(2)}{k}[\/latex]<\/p>\n

                                                  Alternative formula:<\/strong> [latex]A(t) = A_0\\left(\\frac{1}{2}\\right)^{\\frac{t}{T}}[\/latex] where [latex]T[\/latex] is the half-life<\/p>\n

                                                  Use:<\/em> Common in radioactive decay and carbon-14 dating<\/p>\n

                                                  Newton’s Law of Cooling<\/strong><\/p>\n

                                                  Temperature of an object in surrounding air:<\/p>\n

                                                  [latex]T(t) = Ae^{kt} + T_s[\/latex]<\/p>\n

                                                  Where:<\/p>\n

                                                    \n
                                                  • [latex]A[\/latex] = difference between initial and surrounding temperature<\/li>\n
                                                  • [latex]k[\/latex] = continuous rate of cooling (negative)<\/li>\n
                                                  • [latex]T_s[\/latex] = surrounding (ambient) temperature<\/li>\n
                                                  • [latex]t[\/latex] = time<\/li>\n<\/ul>\n

                                                    Strategy:<\/strong><\/p>\n

                                                      \n
                                                    1. Set [latex]T_s[\/latex] equal to ambient temperature<\/li>\n
                                                    2. Use initial conditions to find [latex]A[\/latex]<\/li>\n
                                                    3. Use a second data point to find [latex]k[\/latex]<\/li>\n
                                                    4. Use model to make predictions<\/li>\n<\/ol>\n

                                                      Logistic Growth Model:<\/strong> [latex]f(x) = \\frac{c}{1 + ae^{-bx}}[\/latex]<\/p>\n

                                                      Where:<\/p>\n

                                                        \n
                                                      • [latex]\\frac{c}{1+a}[\/latex] = initial value<\/li>\n
                                                      • [latex]c[\/latex] = carrying capacity (limiting value\/maximum)<\/li>\n
                                                      • [latex]b[\/latex] = constant determined by rate of growth<\/li>\n<\/ul>\n

                                                        Choosing the Right Model<\/strong><\/p>\n

                                                        \n\n\n\n\n\n\n\n
                                                        Model<\/th>\nWhen to Use<\/th>\nConcavity<\/th>\n<\/tr>\n<\/thead>\n
                                                        Exponential<\/strong><\/td>\nRapid growth\/decay without bound<\/td>\nAlways concave up<\/td>\n<\/tr>\n
                                                        Logarithmic<\/strong><\/td>\nRapid change at first, then slows<\/td>\nAlways concave down<\/td>\n<\/tr>\n
                                                        Logistic<\/strong><\/td>\nGrowth with upper limit<\/td>\nChanges from concave up to concave down<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                                        Fitting Exponential Models to Data<\/h3>\n

                                                        Using a Graphing Calculator for Regression:<\/strong><\/p>\n

                                                          \n
                                                        1. Enter data in lists (L1 for input, L2 for output)<\/li>\n
                                                        2. Create scatter plot to identify pattern<\/li>\n
                                                        3. Use appropriate regression command:\n
                                                            \n
                                                          • ExpReg<\/strong> for exponential<\/li>\n
                                                          • LnReg<\/strong> for logarithmic<\/li>\n
                                                          • Logistic<\/strong> for logistic<\/li>\n<\/ul>\n<\/li>\n
                                                          • Graph model with data to verify fit<\/li>\n
                                                          • Check [latex]r^2[\/latex] value (closer to 1 = better fit)<\/li>\n<\/ol>\n

                                                            Interpolation vs. Extrapolation<\/strong><\/p>\n

                                                              \n
                                                            • Interpolation:<\/strong> Predictions within the data range (more reliable)<\/li>\n
                                                            • Extrapolation:<\/strong> Predictions outside the data range (less reliable, requires careful reasoning)<\/li>\n<\/ul>\n

                                                              Key Equations<\/h2>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                                              Product rule for logarithms<\/strong><\/td>\n[latex]\\log_b(MN) = \\log_b(M) + \\log_b(N)[\/latex]<\/td>\n<\/tr>\n
                                                              Quotient rule for logarithms<\/strong><\/td>\n[latex]\\log_b\\left(\\frac{M}{N}\\right) = \\log_b(M) - \\log_b(N)[\/latex]<\/td>\n<\/tr>\n
                                                              Power rule for logarithms<\/strong><\/td>\n[latex]\\log_b(M^n) = n\\log_b(M)[\/latex]<\/td>\n<\/tr>\n
                                                              Change-of-base formula<\/strong><\/td>\n[latex]\\log_b(M) = \\frac{\\log_n(M)}{\\log_n(b)} = \\frac{\\ln(M)}{\\ln(b)}[\/latex]<\/td>\n<\/tr>\n
                                                              One-to-one property (exponential)<\/strong><\/td>\n[latex]b^S = b^T \\Leftrightarrow S = T[\/latex]<\/td>\n<\/tr>\n
                                                              One-to-one property (logarithmic)<\/strong><\/td>\n[latex]\\log_b(S) = \\log_b(T) \\Leftrightarrow S = T[\/latex]<\/td>\n<\/tr>\n
                                                              Continuous growth\/decay<\/strong><\/td>\n[latex]A(t) = A_0 e^{kt}[\/latex]<\/td>\n<\/tr>\n
                                                              Doubling time<\/strong><\/td>\n[latex]t = \\frac{\\ln(2)}{k}[\/latex]<\/td>\n<\/tr>\n
                                                              Half-life<\/strong><\/td>\n[latex]t = -\\frac{\\ln(2)}{k}[\/latex] or [latex]A(t) = A_0\\left(\\frac{1}{2}\\right)^{\\frac{t}{T}}[\/latex]<\/td>\n<\/tr>\n
                                                              Newton’s Law of Cooling<\/strong><\/td>\n[latex]T(t) = Ae^{kt} + T_s[\/latex]<\/td>\n<\/tr>\n
                                                              Logistic growth<\/strong><\/td>\n[latex]f(x) = \\frac{c}{1 + ae^{-bx}}[\/latex]<\/td>\n<\/tr>\n
                                                              Exponential regression<\/strong><\/td>\n[latex]y = ab^x[\/latex]<\/td>\n<\/tr>\n
                                                              Logarithmic regression<\/strong><\/td>\n[latex]y = a + b\\ln(x)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                              Glossary<\/h2>\n

                                                              carrying capacity<\/strong><\/p>\n

                                                              The limiting value [latex]c[\/latex] in a logistic model; represents the maximum sustainable population or quantity.<\/p>\n

                                                              change-of-base formula<\/strong><\/p>\n

                                                              A formula that allows evaluation of logarithms with any base using logarithms of another base: [latex]\\log_b(M) = \\frac{\\log_n(M)}{\\log_n(b)}[\/latex].<\/p>\n

                                                              concavity<\/strong><\/p>\n

                                                              The direction a curve bends; concave up curves bend upward (can hold water), concave down curves bend downward (cannot hold water).<\/p>\n

                                                              continuous growth\/decay model<\/strong><\/p>\n

                                                              A model of the form [latex]A(t) = A_0 e^{kt}[\/latex] where [latex]k > 0[\/latex] represents growth and [latex]k < 0[\/latex] represents decay.<\/p>\n

                                                              extraneous solution<\/strong><\/p>\n

                                                              A solution that emerges algebraically but doesn’t satisfy the original equation; common when logarithms of negative numbers or zero would be required.<\/p>\n

                                                              extrapolation<\/strong><\/p>\n

                                                              Using a model to make predictions outside the range of original data; less reliable and requires careful reasoning.<\/p>\n

                                                              half-life<\/strong><\/p>\n

                                                              The time required for an exponentially decaying quantity to reduce to half its original amount; calculated as [latex]t = -\\frac{\\ln(2)}{k}[\/latex].<\/p>\n

                                                              identity property of logarithms<\/strong><\/p>\n

                                                              The logarithm of the base to itself equals 1: [latex]\\log_b(b) = 1[\/latex].<\/p>\n

                                                              interpolation<\/strong><\/p>\n

                                                              Using a model to make predictions within the range of original data; generally more reliable than extrapolation.<\/p>\n

                                                              inverse property of logarithms<\/strong><\/p>\n

                                                              Logarithms and exponentials undo each other: [latex]b^{\\log_b(x)} = x[\/latex] and [latex]\\log_b(b^x) = x[\/latex].<\/p>\n

                                                              logistic growth<\/strong><\/p>\n

                                                              Growth that is exponential at first but slows as it approaches a maximum value (carrying capacity); modeled by [latex]f(x) = \\frac{c}{1 + ae^{-bx}}[\/latex].<\/p>\n

                                                              Newton’s Law of Cooling<\/strong><\/p>\n

                                                              A model describing how an object’s temperature changes over time to equalize with surrounding temperature: [latex]T(t) = Ae^{kt} + T_s[\/latex].<\/p>\n

                                                              one-to-one property of exponential functions<\/strong><\/p>\n

                                                              If [latex]b^S = b^T[\/latex], then [latex]S = T[\/latex] for any positive real number [latex]b \\neq 1[\/latex].<\/p>\n

                                                              one-to-one property of logarithmic functions<\/strong><\/p>\n

                                                              If [latex]\\log_b(S) = \\log_b(T)[\/latex], then [latex]S = T[\/latex] for positive real numbers [latex]S[\/latex], [latex]T[\/latex], and base [latex]b > 0[\/latex], [latex]b \\neq 1[\/latex].<\/p>\n

                                                              power rule for logarithms<\/strong><\/p>\n

                                                              The logarithm of a power equals the exponent times the logarithm: [latex]\\log_b(M^n) = n\\log_b(M)[\/latex].<\/p>\n

                                                              product rule for logarithms<\/strong><\/p>\n

                                                              The logarithm of a product equals the sum of the logarithms: [latex]\\log_b(MN) = \\log_b(M) + \\log_b(N)[\/latex].<\/p>\n

                                                              property of logarithmic equality<\/strong><\/p>\n

                                                              If [latex]\\log_b(M) = \\log_b(N)[\/latex], then [latex]M = N[\/latex] for any positive real numbers [latex]M[\/latex], [latex]N[\/latex], and base [latex]b > 0[\/latex], [latex]b \\neq 1[\/latex].<\/p>\n

                                                              quotient rule for logarithms<\/strong><\/p>\n

                                                              The logarithm of a quotient equals the difference of the logarithms: [latex]\\log_b\\left(\\frac{M}{N}\\right) = \\log_b(M) - \\log_b(N)[\/latex].<\/p>\n

                                                              radiocarbon dating<\/strong><\/p>\n

                                                              A method for determining the age of organic materials by measuring the ratio of carbon-14 to carbon-12, based on carbon-14’s half-life of 5,730 years.<\/p>\n

                                                              regression<\/strong><\/p>\n

                                                              A method of fitting an algebraic model to data<\/p>\n

                                                              zero property of logarithms<\/strong><\/p>\n

                                                              The logarithm of 1 to any base equals 0: [latex]\\log_b(1) = 0[\/latex].<\/p>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":510,"module-header":"cheat_sheet","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/692"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":11,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/692\/revisions"}],"predecessor-version":[{"id":5373,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/692\/revisions\/5373"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/510"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/692\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=692"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=692"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=692"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=692"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}