Algebra II Chapter 2 Lesson Plans s1

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, x b = y iff x = logb y

Exponential form Logarithmic form: 3 10 = 1000 3 = log10 1000

exponent

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)

x What do you notice? f(x) = 10 and f(x) = log10x are the inverse functions of each other.

x y The inverse of y = 10 is x = 10 , 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: 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

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 eln 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

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)

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)