Advanced Algebra Trig Lesson Plans Section 5.2 – Logarithmic Functions Enduring Understandings: The student shall be able to: 1. Recognize and evaluate logarithmic functions 2. Graph logarithmic functions 3. Use logarithmic functions to solve real life problems Standards: 10 Exponential and Logarithmic Functions Defines, graphs, and shows the inverse relationship between logarithmic and exponential functions 11 Exponential and Logarithmic Functions Solves logarithmic and exponential equations and problems 12 Exponential and Logarithmic Functions Solves problems involving application of exponential and logarithmic functions using appropriate techniques and tools - Makes predictions from collected data using regression techniques Essential Questions: What is a logarithmic function, and how does it differ from an exponential function? How are exponential and logarithmic functions related? Activities: We have been talking about exponential functions. We are now going to study logarithmic functions. Equivalent Exponential and Logarithmic forms: For any positive base b, where b 1, bx = y iff x = logb y Exponential form 103 = 1000 Logarithmic form: 3 = log10 1000 exponent base (emphasize) Fill in the following tables: Complete for f(x) = 10x x -2 f(x) -1 0 1 2 Complete the table for f(x) = log10x x 1/100 1/10 f(x) 1 10 100 What do you notice? f(x) = 10x and f(x) = log10x are the inverse functions of each other. The inverse of y = 10x is x = 10y, which in logarithmic form is y = log10 x. What does inverse mean? Geometrically, they are reflected across the y = x line Algebraically, they “undo” each other, so f(g(x)) = g(f(x)) = x A table can be used to find x for 10^x = 1000, but to find x for 10^x = 2.3 takes a logarithm. Write log as exponential # 2 – 6 even Write exponential as log # 10 – 18 even Properties of logarithms: 1. loga1 = 0 because a0 = 1 2. logaa = 1 because a1 = a 3. log a a x x and a loga x x inverse property 4. If logax = logay then x = y one=-to-one Property The Natural Logarithmic Function is defined by: f(x) = logex = ln x, x > 0 The above Properties of Logarithms applies to natural logs, since e is just a specific base, but they look different: 1. ln 1 = 0 because e0 = 1 2. ln e = 1 because e1 = e 3. ln e x x and e ln x x inverse property 4. If ln x = ln y then x = y one=-to-one Property Evaluate expressions without a calculator # 20 – 26 even Use a calculator to solve expressions # 34 – 40 even Describe the graph if b = 1 in y = logb x. (the equivalent form is x = by, so if b = 1, we would have x = 1 for an equation, i.e., a vertical line) Is y = log1 x a function? (no – it does not pass the vertical line test) Find the domain and x-intercept and vertical asymptote of log function: # 52 – 62 even Applications: Assessments: Hw pg 399 – 402, # 1 – 61 by 4’s, 75, 77, 79, 83, 85 (21)