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1 Exponential Functions and Logarithmic Functions Standards 11, 13, 14, 15 Using Common Logarithms with other bases than 10 More exponential equations Common Logarithm or base 10 logarithms Solving Exponential Equations Logarithmic Functions: Comparing to exponential Exponential Functions: Introduction Logarithmic Equations Natural Logarithms END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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2 STANDARD 11: Students prove simple laws of logarithms. 11.1 Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. 11.2 Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step STANDARD 13: Students use the definition of logarithms to translate between logarithms in any base. STANDARD 14 Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values. STANDARD 15: Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is some-times true, always true, or never true. ALGEBRA II STANDARDS THIS LESSON AIMS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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3 ESTÁNDAR 11: Los estudiantes prueban simples leyes de logaritmos. 11.1 Los estudiantes entienden la relación inversa entre exponentes y logaritmos y usan esta relación para resolver problemas que involucran logaritmos y exponentes. 11.2 Los estudiantes juzgan la validez de un argumento de acuerdo a si las propiedades de los números reales, exponentes y logaritmos han sido correctamente aplicados en cada paso. ESTÁNDAR 13: Los estudiantes usan la definición de logaritmos para convertir de una base a otra base logaritmos. ESTÁNDAR 14: Los estudiantes entienden y usan las propiedades de logaritmos para simplificar expresiones numéricas logarítmicas e identificar sus valores aproximados apropiados. ESTÁNDAR 15: Los estudiantes determinan si un estatuto algebraico que involucra expresiones racionales, expresiones radicales, o logaritmos o funciones exponenciales es algunas veces cierto, siempre cierto o nunca verdadero. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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4 Standards 11, 13, 14, 15 Exponential Functions: 4 2 6 -2-4-6 2 4 6 -2 -4 -6 8 10 -8 -10 8 -8 10 x y y= 2 x DEFINITION OF EXPONENTIAL FUNCTION: An equation of the form, where a = 0, b > 0, and b = 1, is called an exponential function with base b. y = a b x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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5 Standards 11, 13, 14, 15 Simplify the following expressions: 3 2 32 = 3 2 = 3 2 32 = 3 64 = 3 8 = 6561 4(5 )(5 ) 2 3 - 4(5 ) 2 3 + - = 3 15 3 3 13 3 3 15 3 = 3 13 3 - = 3 2 3 4 = 4(5) = 0 1 = (3) 2 3 = 9 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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6 Standards 11, 13, 14, 15 Find the value a if the graph of an exponential function of the form passes through the given point: A(3, 45) y = a 4 x x 45 = a 4 3 45 = a 64 64 a= 45 64 B(2, 64) y = a 4 x 64 = a 4 2 64 = a 16 16 a= 4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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7 Standards 11, 13, 14, 15 Solve the following exponential equation or inequality: 16 4 2x + 1 2x + 12 = 2x + 1 2x + 12 = 2 4 2 2 2 2 4(2x + 1) 2(2x + 12) = 4(2x+1) = 2(2x + 12) 8x + 4 = 4x + 24 -4 8x = 4x + 20 -4x 4x = 20 4 x = 5 27 9 4x - 2 3x + 9 < 4x - 2 3x +9 < 3 3 3 2 3 3 3(4x - 2) 2(3x + 9) < 3(4x- 2) < 2(3x + 9) 12x - 6< 6x + 18 +6 12x < 6x + 24 -6x 6x <24 6 x < 4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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Standards 11, 13, 14, 15 x2 = yy 2 = y 2 3 6 y2 = xx 2 = 0.50.5 2 = 44 2 = 88 2 = 6464 x 3 2 6 y y y y 0.5 4 64 8 2 3 6 y 2 = y x 2 = x y Getting the INVERSE for the exponential function: Log x = y 2 Solving for y: y=x x y y= 2 x Log x = y 2 (0,1) (1,0) LOGARITHMIC FUNCTIONS So the logarithmic functions and exponential functions are inverse one from the other! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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9 Standards 11, 13, 14, 15 n = b p p = log n b Exponential Equation Logarithmic Equation number 81 = 3 4 4 = log 81 3 125 = 5 3 3 = log 125 5 279936 = 6 7 7 = log 279936 6 DEFINITION OF LOGARITHM Suppose b > 0 and b = 1. For n > 0, there is a number p such that log n=p if and only if b = n. b p base exponent or logarithm PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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10 Standards 11, 13, 14, 15 Solve the following logarithmic equations: x = log 243 3 3 3 = 243 x 3 = 3 x 5 x = 5 5 = log 7776 b b b = 7776 5 b = 6 5 5 4 = log n 2 2 n = 2 4 n = 16 Log (6x + 2) = log (3x +8) 6 6 6x + 2 = 3x + 8 -2 6x = 3x + 6 -3x 3x = 6 3 x = 2 Suppose b > 0 and b=1. Then log x = log x if and only if x = x b b 1 2 12 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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11 Standards 11, 13, 14, 15 Evaluate each expression: log 128 2 = log 2 2 7 log b = x b x = 7 8 log (16) 8 = 16 b log x b = x y=x x y y= b x Log x = y b (0,1) (1,0) Remember that exponential and logarithmic functions are mutually inverse! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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12 Standards 11, 13, 14, 15 4log 3 + 2log 5 = log x 7 77 log 3 + log 5 = log x 777 4 2 log 81 + log 25 = log x 777 log (81)(25) = log x 7 7 log (2025) = log x 7 7 x = 2025 with b=1 log m = p log m b p b Suppose b > 0 and b=1. Then log x = log x if and only if x = x b b 1 2 12 log mn = log m + log n b b b with b=1 Solve 4log 3 + 2log 5 = log x: 7 77 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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13 Standards 11, 13, 14, 15 6log 2 - 2log 4 = log x 5 55 log 2 - log 4 = log x 555 6 2 log 64 - log 16 = log x 555 log (4) = log x 5 5 x = 4 with b=1 log m = p log m b p b Suppose b > 0 and b=1. Then log x = log x if and only if x = x b b 1 2 12 Solve 6log 2 - 2log 4 = log x: 5 55 log = log x 5 5 64 16 log = log m - log n b b b with b=1 m n PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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14 Standards 11, 13, 14, 15 n =10 p p = log n 10 number base exponent or logarithm Exponential Equation Logarithmic Equation COMMON LOGARITHM: LOGARITHM WITH BASE 10 p = log n Most Calculators only have COMMON LOGARITHM or Logarithm with base 10! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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15 Standards 11, 13, 14, 15 Find the value for each logarithm and state the characteristic and mantissa: Log 0.0008= -3.09691 -3.09691 + 10 - 10 6.90309 - 10 6-10= -4 6.90309 -6=0.90309 Characteristic Mantissa To find the characteristic and mantissa of a negative logarithm, it is necessary to express the exponent as a sum of an integer and a positive decimal Log 69.8 = 1.84386 1 Characteristic 0.84386 Mantissa PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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16 Standards 11, 13, 14, 15 Calculating logarithms with a base other than 10 using Common Logarithm: log n = a log n Log a b b if b=10 then log n = a log n Log a log 130 2 log 2 = = 2.11394 0.30103 = 7.02237 log 210 6 log 6 = = 2.32222 0.77815 = 2.98428 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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17 Standards 11, 13, 14, 15 Solve the following exponential equation: 5 7x -3 8 4x + 5 = log 8 = log 5 7x -3 4x + 5 (4x + 5)log 8 = (7x-3)log5 4x log 8 + 5 log 8 = 7x log 5 -3 log 5 -5 log 8 4x log 8 = 7x log 5 -3 log 5 - 5 log 8 -7x log 5 4x log 8 – 7x log 5 = -3 log 5 – 5 log 8 x(4log 8 – 7 log 5) = -3 log 5 – 5 log8 4 log 8 – 7 log 5 x = -3(.69897 )- 5(.90309) 4(.90309) - 7(.69897) x= 5.16 with b=1 log m = p log m b p b This method is useful when the base of the exponential expressions can’t be equal! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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18 Standards 11, 13, 14, 15 n =e p p = log n e number base exponent or logarithm Exponential Equation Logarithmic Equation NATURAL LOGARITHM: LOGARITHM WITH BASE e p = LN e= 2.718281828459 It is important to observe that the Exponential Function and the Natural Logarithm Functions are inverses one from the other. NATURAL LOGARITHM Most calculators have them as: e x and LN PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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