L9 - Exponential and Logarithmic Functions PDF

BASIC MATHEMATICS (FADS 1004) BY: MDM FAZILAH AHMAD © 2019, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author. EXPONENTIAL Exponential Functions and...

BASIC MATHEMATICS (FADS 1004) BY: MDM FAZILAH AHMAD © 2019, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author. EXPONENTIAL Exponential Functions and graphs AND Logarithmic Functions and graphs LOGARITHMIC SOLVING Exponential and FUNCTIONS Logarithmic Equations Exponential Functions Definition: The exponential functions f with base a is denoted by f ( x) = a x where a is positive constant other than 1 (a > 0). Example: Exponential Functions f ( x) = 2 x x y=2x x y=2x x y=2x 0 1 -1 0.5 -2 0.25 1 2 -2 0.25 -1 0.5 2 4 -3 0.125 0 1 3 8 -4 0.06 1 2 4 16 -5 0.03 2 4 5 32 3 8 positive x-values negative x-values negative to positive x-values Exponential Functions x y=2x -2 0.25 -1 0.5 0 1 1 2 2 4 3 8  Exponential Functions The characteristics of Exponential Functions: 1. If a > 1, y = a xhas a graph that goes up to the right and increasing functions. 2. If 0 < a < 1, y = a has a graph that goes down to the right and x decreasing functions. Graph of Exponential Functions 1 y = 3− x and y = ( ) x y = 3x y = 2x 3 Transformations of Graphs of Exponential Functions :REFLECTION g ( x) = −b x g ( x) = 2− x g ( x) = 2 x g ( x) = b − x g ( x ) = −2 x Transformations of Graphs of Exponential Functions VERTICAL SHIFTING g ( x) = b + c x g ( x) = b − c x 2x + 1 x 2 2 −1 x Transformations of Graphs of Exponential Functions:HORIZONTAL SHIFT g ( x) = b x +c g ( x) = 2 x g ( x) = b x −c g ( x) = 2 x +1 g ( x) = 2 x −1 Graphing Exercise 𝑦 = 2𝑥 𝑦 = 5𝑥 𝑦 = 3𝑥 + 1, 𝑦 = 3𝑥 − 1 𝑦 = 3𝑥+1 , 𝑦 = 3𝑥−2 13 Graphing Exercise 𝑦 = 2𝑥 𝑦 = 5𝑥 𝑦 = 3𝑥 + 1, 𝑦 = 3𝑥 − 1 𝑦 = 3𝑥+1 , 𝑦 = 3𝑥−2 14  The Natural Base e The exponential function with base e is called the natural exponential function. e has a value attached it (similar to pi). e is approximately 2.718281828... This base is used in economic analysis and problems involving natural growth and decay. At this point, we are just going to learn how to find the value of e raised to an exponent using the calculator. The Natural Base e The natural exponential function is the exponential function with base e. It is often referred to as the exponential function. Since 2 < e < 3, the graph of the natural exponential function lies between the graphs of y = 2x and y = 3x, as shown below.  Exponential Functions EXPONENTIAL Logarithmic Functions AND Properties of Logarithmic Functions LOGARITHMIC SOLVING Exponential and Logarithmic Equations FUNCTIONS EXPONENTIAL AND LOGARITHMIC MODELS  Recognize Recognize and evaluate a logarithmic functions with and evaluate base a. Sketch Sketch the graph of a logarithmic function. Learning Outcomes Recognize and Recognize and evaluate the natural logarithmic function. evaluate Use a logarithmic model to solve an application Use problem.  Logarithmic Functions Every function of the form f (x) = ax passes the Horizontal Line Test and therefore must have an inverse function. This inverse function is called the logarithmic function with base a.  The equations y = loga x and x = ay are equivalent. The first equation is in logarithmic form and the second is in exponential form. Logarithmic For example, the logarithmic equation Functions 2 = log3 9 can be rewritten in exponential form as 9 = 32. The exponential equation 53 = 125 can be rewritten in logarithmic form as log5 125 = 3.  Logarithmic Functions The logarithmic function with base 10 is called the common logarithmic function. It is denoted by log10 or simply by log. The following properties follow directly from the definition of the logarithmic function with base a. Example: a ) log 7 7 = 1 b) log 5 1 = 0  Graphs of Logarithmic Functions The basic characteristics of logarithmic graphs Graph of y = loga x, a  1 Domain: (0, ) Range: ( , ) x-intercept: (1, 0) Increasing Vertical Asymptote, x=0 Transformations of Graphs of Logarithmic Functions 𝑦 = log 2 𝑥 𝑦 = log 3 𝑥 40 Transformations of Graphs of Logarithmic Functions VERTICAL SHIFTING 𝑦 = log 2 𝑥 + 1 𝑦 = log 2 𝑥 𝑦 = log 2 𝑥 − 1 𝑦 = 𝑙𝑜𝑔 𝑥 + C 𝑐 > 0, 𝑔𝑟𝑎𝑝ℎ𝑠 𝑔𝑜 𝑢𝑝𝑤𝑎𝑟𝑑 𝑏𝑦 𝑐 𝑢𝑛𝑖𝑡 𝑐 < 0, 𝑔𝑟𝑎𝑝ℎ𝑠 𝑔𝑜 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑 𝑏𝑦 𝑐 𝑢𝑛𝑖𝑡 [email protected] 41  Transformations of Graphs of Logarithmic Functions HORIZONTAL SHIFTING 𝑦 = log 2 (𝑥 + 1) 𝑦 = 𝑙𝑜𝑔(𝑥 + c) 𝑐 > 0, 𝑔𝑟𝑎𝑝ℎ𝑠 𝑔𝑜 𝑙𝑒𝑓𝑡 𝑏𝑦 𝑐 𝑢𝑛𝑖𝑡 𝑦 = log 2 (𝑥 − 1) 𝑐 < 0, 𝑔𝑟𝑎𝑝ℎ𝑠 𝑔𝑜 𝑟𝑖𝑔ℎ𝑡 𝑏𝑦 𝑐 𝑢𝑛𝑖𝑡 𝑦 = log 2 𝑥 [email protected] 42 Transformations of Graphs of Logarithmic Functions REFLECTION 𝑦 = log 2 𝑥 𝑦 = −log 2 𝑥 𝑦 = −𝑙𝑜𝑔 𝑥 𝑅𝑒𝑓𝑙𝑒𝑐𝑡 𝑡𝑜 𝑥 − 𝑎𝑥𝑖𝑠 [email protected] 43 Example, 𝑦 = −log10 𝑥 − 1 + 2 Vertical Shifting, 𝑦 = 𝑙𝑜𝑔 𝑥 + C 𝑐 > 0, 𝑔𝑟𝑎𝑝ℎ𝑠 𝑔𝑜 𝑢𝑝𝑤𝑎𝑟𝑑 𝑏𝑦 𝑐 𝑢𝑛𝑖𝑡 𝑐 < 0, 𝑔𝑟𝑎𝑝ℎ𝑠 𝑔𝑜 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑 𝑏𝑦 𝑐 𝑢𝑛𝑖𝑡 Horizontal Shifting, 𝑦 = 𝑙𝑜𝑔(𝑥 + c) 𝑐 > 0, 𝑔𝑟𝑎𝑝ℎ𝑠 𝑔𝑜 𝑙𝑒𝑓𝑡 𝑏𝑦 𝑐 𝑢𝑛𝑖𝑡 𝑐 < 0, 𝑔𝑟𝑎𝑝ℎ𝑠 𝑔𝑜 𝑟𝑖𝑔ℎ𝑡 𝑏𝑦 𝑐 𝑢𝑛𝑖𝑡 Reflection, 𝑦 = −𝑙𝑜𝑔 𝑥 𝑅𝑒𝑓𝑙𝑒𝑐𝑡 𝑡𝑜 𝑥 − 𝑎𝑥𝑖𝑠 44  The Natural Logarithmic Function We will see that f (x) = ex is one-to-one and so has an inverse function. This inverse function is called the natural logarithmic function and is denoted by the special symbol ln x, read as “the natural log of x” The Natural Logarithmic Function The four properties of logarithms are also valid for natural logarithms.  The Natural Logarithmic Function Graph of y = In x Reflection of graph of f (x) = ex about the line y = x. Learning Outcomes: Properties of Logarithmic function ✓Evaluate a logarithm using the change of base formula. ✓Use properties of logarithms to evaluate or rewrite a logarithmic expression. ✓Use properties of logarithms to expand or condense a logarithmic expression. ✓Use logarithmic functions to model and solve real life applications.  Change of Base Most calculators have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base e). Although common logarithms and natural logarithms are the most frequently used, you may occasionally need to evaluate logarithms with other bases. To do this, you can use the following change-of-base formula.  Example – Changing Bases Using Common Logarithms a) Use a calculator. Simplify. b) Properties of Logarithms  Example: Properties of Log Simplify each logarithmic expression. 1. log4 5 + log4 𝑥3 + log4 𝑦 2. log 3 𝑥 2 + log 3 13 − log 3 2𝑦 54  Example– Expanding Logarithmic Expressions Expand each logarithmic expression. a. log4 5x3y b. Solution: a. log4 5x3y = log4 5 + log4 x3 + log4 y Product Property = log4 5 + 3 log4 x + log4 y Power Property Example – Solution (b) cont’d Rewrite using rational exponent. Quotient Property Power Property  Exponential Functions and graphs EXPONENTIAL Logarithmic Functions and graphs AND LOGARITHMIC SOLVING Exponential and Logarithmic Equations FUNCTIONS Application of exponential and Logarithmic functions Learning Outcomes ✓Solve an exponential equation. ✓Solve a logarithmic equation. ✓Use an exponential or a logarithmic model to solve an application problem. Solving Exponential Equations Solve each equation and approximate the result to three decimal places, if necessary. 2 a. e –x = e – 3x – 4 b. 3(2x) = 42 Example – Solution (a) 2 e –x = e – 3x – 4 Write original equation. –x2 = –3x – 4 One-to-One Property x2 – 3x – 4 = 0 Write in general form. (x + 1)(x – 4) = 0 Factor. (x + 1) = 0 x = –1 Set 1st factor equal to 0. (x – 4) = 0 x=4 Set 2nd factor equal to 0. The solutions are x = –1 and x = 4. Check these in the original equation.  Example– Solution (b) cont’d 3(2x) = 42 Write original equation. 2x = 14 Divide each side by 3. log 2x = log 14 Take log (base 10) of each side. x log 2 = log 14 Inverse Property log 14 x= Change-of-base formula log 2  3.807 The solution is x = log2 14  3.807. Check this in the original equation.  Solving Logarithmic Equations To solve a logarithmic equation, you can write it in exponential form. Logarithmic form ln x = 3 Exponentiate each side. eln x = e3 Exponential form x = e3 This procedure is called exponentiating each side of an equation. Example– Solving Logarithmic Equations Original equation a. ln x = 2 Exponentiate each side. eln x = e2 Inverse Property x = e2 b. log3(5x – 1) = log3(x + 7) Original equation One-to-One Property 5x – 1 = x + 7 Add –x and 1 to each side. 4x = 8 Divide each side by 4. x=2  Example– Solving Logarithmic Equations cont’d c. log6(3x + 14) – log6 5 = log6 2x Original equation Quotient Property of Logarithms One-to-One Property 3x + 14 = 10x Cross multiply. –7x = –14 Isolate x. x=2 Divide each side by –7. Thank you Address Telephone Website University of Cyberjaya 03 - 8313 7000 www.cyberjaya.edu.my Persiaran Bestari, Cyber 11, 63000 Cyberjaya, Facsimile Email Selangor Darul Ehsan, Malaysia. 03 – 8313 7001 [email protected]