<p> Advanced Algebra Trig Lesson Plans</p><p>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</p><p>Standards: 10 Exponential and Logarithmic Functions Defines, graphs, and shows the inverse relationship between logarithmic and exponential functions</p><p>11 Exponential and Logarithmic Functions Solves logarithmic and exponential equations and problems</p><p>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</p><p>Essential Questions:  What is a logarithmic function, and how does it differ from an exponential function?  How are exponential and logarithmic functions related?</p><p>Activities: We have been talking about exponential functions. We are now going to study logarithmic functions. </p><p>Equivalent Exponential and Logarithmic forms: For any positive base b, where b  1, x b = y iff x = logb y</p><p>Exponential form Logarithmic form: 3 10 = 1000 3 = log10 1000</p><p> exponent</p><p> base (emphasize) Fill in the following tables: Complete for f(x) = 10x x -2 -1 0 1 2 f(x) Complete the table for f(x) = log10x x 1/100 1/10 1 10 100 f(x)</p><p> x What do you notice? f(x) = 10 and f(x) = log10x are the inverse functions of each other. </p><p> x y The inverse of y = 10 is x = 10 , which in logarithmic form is y = log10 x. </p><p>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</p><p>A table can be used to find x for 10^x = 1000, but to find x for 10^x = 2.3 takes a logarithm. </p><p>Write log as exponential # 2 – 6 even Write exponential as log # 10 – 18 even</p><p>Properties of logarithms: 0 1. loga1 = 0 because a = 1 1 2. logaa = 1 because a = a x loga x 3. log a a  x and a  x inverse property</p><p>4. If logax = logay then x = y one=-to-one Property</p><p>The Natural Logarithmic Function is defined by: f(x) = logex = ln x, x > 0</p><p>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 eln x  x inverse property 4. If ln x = ln y then x = y one=-to-one Property</p><p>Evaluate expressions without a calculator # 20 – 26 even Use a calculator to solve expressions # 34 – 40 even</p><p> y Describe the graph if b = 1 in y = logb x. (the equivalent form is x = b , 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)</p><p>Find the domain and x-intercept and vertical asymptote of log function: # 52 – 62 even</p><p>Applications: </p><p>Assessments: Hw pg 399 – 402, # 1 – 61 by 4’s, 75, 77, 79, 83, 85 (21)</p>