Algebra II Page 1! of 11 ! 4-3 Study Guide Logarithmic Functions ! Attendance Problems. Use mental math (no calculator, do all of your work mentally) to evaluate the following. 1. ! 4 −3 2. ! 16 1 4 3. ! 10 −5 ⎛ 2⎞ 4. ! ⎜ ⎟ ⎝ 3⎠ −3 ! ! 2. A power has a base of –2 and exponent of 4. Write and evaluate the power. ! ! ! ! ! • I can write equivalent forms for exponential and logarithmic functions. • I can write, evaluate, and graph logarithmic functions. ! Vocabulary logarithm common logarithm logarithmic function Common Core • CCSS.MATH.CONTENT.HSA.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. • CCSS.MATH.CONTENT.HSA.CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. • CCSS.MATH.CONTENT.HSF.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* ! ! Algebra II 4-3 Study Guide Page 2! of 11 ! How many times would you have to double $1 before you had $8? You could use an exponential equation to model this situation. 1(2x) = 8. You may be able to solve this equation by using mental math if you know 23 = 8. So you would have to double the dollar 3 times to have $8. ! How many times would you have to double $1 before you had $512? You could solve this problem if you could solve 2x = 8 by using an inverse operation that undoes raising a base to an exponent equation to model this situation. This operation is called finding the logarithm. A logarithm is the exponent to which a specified base is raised to obtain a given value. ! You can write an exponential equation as a logarithmic equation and vice versa. Video Example 1. Write each exponential equation in logarithmic form. A. ! 5 3 = 125 ! ! D. ! 10 ! ! −2 = 0.01 B. ! 61 = 6 E. ! 4 x = 16 C. ! 9 0 = 1 E b4-3 > 0, b ≠ 1Guide Study Algebra II 1 Page 3! of 11 ! Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form a. 2 6 = 64 log 2 64 = 6 b. 41 = 4 log 4 4 = 1 The exponent is the logarithm. c. 50 = 1 log 5 1 = 0 Any nonzero base to the 0 power is 1. d. 5 -2 = 0.04 log 5 0.04 = -2 e. 3 x = 81 log 3 81 = x The base of the exponent becomes the base of the logarithm. An exponent (or log) can be negative. The log (and the exponent) can be a variable. Write each exponential equation in logarithmic form. Example 1. Write in logarithmic0 form. 2 each exponential equation 3 ! ! A.! 35 = 2431a. D. ! 6 −1 = ! ! 1 6 9 = 81 B. ! 10 0 =1b. 3 = 27 1 E. ! a b = c x = 4 C. ! 101c. = 10,000 1(x ≠ 0) 4-3 Logarithmic Functions 249 Guided Practice. Write each exponential equation in logarithmic form. ! ! 3. ! 9 2 = 81 4. ! 33 = 27 5. ! x 0 = 1 ( x ≠ 0 ) Video Example 2. Write each logarithmic form in exponential equation. • ! log 5 25 = 2 ! ! D. ! log 3 = 1 ! ! 3 B. ! log 2 8 = 3 E. ! log 8 1 = 0 C. ! log 5 0.2 = −1 5/4/11 3:18:1 LE Algebra II 2 Page 4! of 11 ! 4-3 Study Guide Converting from Logarithmic to Exponential Form Write each logarithmic equation in exponential form. Logarithmic Equation a. log 10 100 = 2 b. log 7 49 = 2 c. Exponential Form The base of the logarithm becomes the base of the power. 10 2 = 100 7 2 = 49 log 8 0.125 = -1 The logarithm is the exponent. A logarithm can be a negative number. 8 -1 = 0.125 d. log 5 5 = 1 51 = 5 e. log 12 1 = 0 12 0 = 1 Example 2. Write logarithmic formequation in exponential equation. form. Writeeach each logarithmic in exponential ! ! A. ! log 9 9 = 1 D. ! log 4 ! ! B. ! log 2 512 = 9 2a. log 10 10 = 1 -3 1 = −2 16 C. ! log 8 64 = 2 2b. log 12 144 = 2 2c. log _1 8 = 2 E. ! log b 1 = 0 A logarithm is an exponent, so the rules for exponents also apply to logarithms. You may have noticed the following properties in the last example. Guided Practice. Write each logarithmic form in exponential equation. 6. ! log10 10 = 1 7. ! log12 144 = 2 Special Properties of Logarithms ! ! 8. ! log 1 8 = −3 2 For any base b such that b > 0 and b ≠ 1, A logarithm is an exponent, so the rules for exponents also apply to logarithms. LOGARITHMIC EXPONENTIAL FORM EXAMPLE You may have noticedFORM the following properties in the last example. Logarithm of Base b ! ! ! ! log bb = 1 b1 = b log 1010 = 1 10 1 = 10 b0 = 1 log 101 = 0 10 0 = 1 Logarithm of 1 log b1 = 0 Special ofStudy Logarithms AlgebraProperties II 4-3 Guide Page 5! of 11 ! For any base b such that b > 0 and b ≠ 1, LOGARITHMIC FORM EXPONENTIAL FORM E b1 = b lo 1 b0 = 1 lo 1 Logarithm of Base b log bb = 1 Logarithm of 1 log b1 = 0 A logarithm with base 10 is called a common logarithm. If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log105. ! A logarithm withmath base 10 is some called a common logarithm . If no b You can use mental to evaluate logarithms. ! a logarithm, the base is assumed to be 10. For example, log 5 = l Video Example 3. Evaluate by using mental math. A. log 100 B. ! log 3 1 9 You can use mental math to evaluate some logarithms. 3 Evaluating Logarithms by Using Mental Math Evaluate by using mental math. A log 1000 1 B log 4 _ 10 ? = 1000 The log is the exponent. 10 3 = 1000 Think: What power of the base is the value? log 1000 = 3 4 1 4? = _ 4 1 4 -1 = _ 4 1 = -1 log 4 _ 4 Algebra II Page 6! of 11 ! 4-3 Study Guide Example 3. Evaluate by using mental math. log 0.01 B. ! log 5 125 C. ! log 5 ! ! 1 5 Guided Practice. Evaluate by using mental math. 9. log 0.00001 10. ! log 25 0.04 ! Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2x, is a logarithmic function, such as y = log2x. ! You may notice that the domain and range of each function are switched. ! The domain of y = 2x is all real numbers (R), and the range is {y|y > 0}. The domain of y = log2x is {x|x > 0}, and the range is all real numbers (R). ! Video Example 4. Use x = -2, -1, 0, 1, and 2 to graph the function ! f ( x ) = 4 x. Then graph its inverse. Describe the domain and the range of the inverse function. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! E The domain of y = 2 is all real numbers (!), and ⎧ ⎫ the range is ⎨ y | y > 0 ⎬. The domain of y = log 2 x is ⎧ ⎫ ⎩ ⎭ > 0 ⎬, andIIthe range is all real (!). ⎨ x | x Algebra 4-3numbers Study Guide ⎩ ⎭ 4 -4 4 8 12 -4 Page 7! of 11 ! Graphing Logarithmic Functions Use the given x-values to graph each function. Then graph its inverse. Describe the domain and range of the inverse function. A f (x) = 3 x; x = -2, -1, 0, 1, and 2 Graph f (x) = 3 x by using a table of values. x -2 -1 1 _ 9 f(x) = 3 x 1 _ 3 0 1 2 1 3 9 y 8 1 _ 9 f -1(x) = log 3 x f(x) = 3x 6 To graph the inverse, f -1(x) = log 3 x, reverse each ordered pair. x (2, 9) 4 f -1(x) = log3 x 2 1 _ 3 1 3 9 -2 -1 0 1 2 (9, 2) -2 2 4 6 x 8 -2 ⎧ ⎫ The domain of f -1(x) is ⎨ x | x > 0 ⎬, and the range is !. ⎩ ⎭ B f (x) = 0.8 x; x = -3, 0, 1, 4, and 7 Graph f (x) = 0.8 x by using a table of values. Round the output values to the nearest tenth, if necessary. x -3 0 f(x) = 0.8 x 2 1 1 4 8 7 y -1 f (x) = log 0.8 x 6 0.8 0.4 0.2 4 -1 To graph f (x) = log 0.8 x, reverse each ordered pair. x 2 f -1(x) = log 0.8 x -3 1 0 0.8 0.4 0.2 1 4 7 2 f(x) = 0.8 x -2 2 4 -2 ⎧ ⎫ The domain of f -1(x) is ⎨ x | x > 0 ⎬, and the range is !. ⎩ ⎭ x () 4. Use x = -2, -1, 1, 2, and 3 to graph f (x) = __34 . Then graph its inverse. Describe the domain and range of the 6 x 8 Algebra II 4-3 Study Guide Example 2. Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function. A. ! f ( x ) = 1.25 x ! ! ! ! ! ! ! ! 1 B. ! f ( x ) = ⎛⎜ ⎞⎟ ⎝ 2⎠ ! x Page 8! of 11 ! Algebra II 4-3 Study Guide Page 9! of 11 ! 11. Guided Practice. Use x = –2, –1, 1, 2, and 3 to x 3 graph ! f ( x ) = ⎛⎜ ⎞⎟ . Then graph its inverse. Describe ⎝ 4⎠ the domain and range of the inverse function. ! ! ! ! ! ! ! Helpful Hint The key is used to evaluate logarithms in base 10. inverse of log. ! is used to find 10x, the Algebra II PLE se s the 5 Page 10 ! of 11 ! 4-3 Study Guide Environmental Application Chemists regularly test rain samples to determine the rain’s acidity, or concentration of hydrogen ions (H +). Acidity is measured in pH, as given by the function pH = log ⎡⎣ H + ⎤⎦, where ⎣⎡ H + ⎤⎦ represents the hydrogen ion concentration in moles per liter. Hydrogen Ion Concentration of Rainwater Central North Dakota 0.0000009 mol/L Eastern Ohio 0.0000629 mol/L Central California 0.0000032 mol/L Find the pH of rainwater from each location. Central New Jersey 0.0000316 mol/L Eastern Texas 0.0000192 mol/L A Central New Jersey The hydrogen ion concentration is 0.0000316 moles per liter. pH = - log ⎡⎣H +⎤⎦ pH = - log (0.0000316) Substitute the known values in the function. Use a calculator to find the value of the key. logarithm in base 10. Press the The rainwater has a pH of about 4.5. B Central North Dakota The hydrogen ion concentration is 0.0000009 moles per liter. pH = - log ⎡⎣H +⎤⎦ pH = - log (0.0000009) Substitute the known values in the function. Use a calculator to find the value of the key. logarithm in base 10. Press the The rainwater has a pH of about 6.0. 5. What is the pH of iced tea with a hydrogen ion concentration of 0.000158 moles per liter? Algebra II 4-3 Study Guide Example 5. The table lists the hydrogen ion concentrations for a number of food items. Find the pH of each. ! ! ! ! ! ! ! ! Substance Page !11 of 11 ! H+ conc. (mol/L) Milk 0.00000025 Tomatoes 0.0000316 Lemon juice 0.0063 12. Guided Practice. What is the pH of iced tea with a hydrogen ion concentration of 0.000158 moles per liter? ! ! 4-3 Assignment (p 253) 17-28; 39-42.