1Wednesday, May 10, Section 4.5 Linearization

Slides:

Advertisements

Similar presentations

Unit 6. For x  0 and 0  a  1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic function with base.

Advertisements

Solving Exponential Equations Equations with variables in exponents, such as 3 x = 5 and 7 3x = 90 are called exponential equations. In Section 9.3, we.

Macroeconomic Facts Chapter 3. 2 Introduction Two kinds of regularities in economic data: -Relationships between the growth components in different variables.

Table of Contents Solving Logarithmic Equations A logarithmic equation is an equation with an expression that contains the log of a variable expression.

Functions and Logarithms

Transforming to Achieve Linearity

Transformations to Achieve Linearity

Exponential and Logarithmic Functions Logarithmic Functions EXPONENTIAL AND LOGARITHMIC FUNCTIONS Objectives Graph logarithmic functions. Evaluate.

5.4 Exponential and Logarithmic Equations Essential Questions: How do we solve exponential and logarithmic equations?

1 Logarithms Definition If y = a x then x = log a y For log 10 x use the log button. For log e x use the ln button.

Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10.

MAC 1105 Section 4.3 Logarithmic Functions. The Inverse of a Exponential Function 

and Logarithmic Equations

Solving Exponential Equations…

Chapter 2 Functions and Graphs Section 6 Logarithmic Functions (Part I)

Logarithmic and Exponential Equations

Table of Contents Solving Exponential Equations An exponential equation is an equation with a variable as part of an exponent. The following examples will.

4.1 Modeling Nonlinear Data.  Create scatter plots of non linear data  Transform nonlinear data to use for prediction.

Section 5.4 Exponential and Logarithmic Models.

Jeopardy 100 Condense Expand Simplify Solve Exponential Solve Logs 500.

Section 4-1: Introduction to Linear Systems. To understand and solve linear systems.

Transcendental Functions Chapter 6. For x  0 and 0  a  1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic.

Logarithms Definition Graphing. What’s a Log? the logarithm of a number to a given base is the power or exponent to which the base must be raised in order.

4.1 Modeling Nonlinear Data.  Create scatter plots of non linear data  Transform nonlinear data to use for prediction  Create residual plots.

How are you all doing? Any questions about anything?

Logarithms 1 Converting from Logarithmic Form to Exponential Form and Back 2 Solving Logarithmic Equations & Inequalities 3 Practice Problems.

11.4 Properties of Logarithms. Logarithms A logarithm is an operation, a little like taking the sine of an angle. Raising a constant to a power is called.

NATURAL LOGARITHMS. The Constant: e e is a constant very similar to π. Π = … e = … Because it is a fixed number we can find e 2.

5.4 Properties of Logarithms 3/1/2013

Solving Exponential Equations. We can solve exponential equations using logarithms. By converting to a logarithm, we can move the variable from the exponent.

( ) EXAMPLE 5 Use inverse properties Simplify the expression. a.

AP STATISTICS Section 4.1 Transforming to Achieve Linearity.

Chapter 8 Systems of Linear Equations in Two Variables Section 8.3.

Solving Equations Involving Logarithmic and Exponential Functions

1 Basic Financial or Economic Data Handling. 2 Changing Indices and Base Changing Frequency (stock vs. flow var.) Nominal vs. Real Logs (linearizing trend.

Chapter 10 Notes AP Statistics. Re-expressing Data We cannot use a linear model unless the relationship between the two variables is linear. If the relationship.

LOGARITHMIC AND EXPONENTIAL EQUATIONS LOGARITHMIC AND EXPONENTIAL EQUATIONS SECTION 4.6.

Goals:  Understand logarithms as the inverse of exponents  Convert between exponential and logarithmic forms  Evaluate logarithmic functions.

Section Logs as Inverse Exponentials. Lesson Objective: Students will: Redevelop the log function by reversing a table for an exponential function.

ConcepTest • Section 1.4 • Question 1

Section 3.4 Solving Exponential and Logarithmic Equations

3.4 Quick Review Express In 56 in terms of ln 2 and ln 7.

MATH 2311 Section 5.5.

Bellwork Find the value of x in each exponential equation.

Bell Ringer Make a scatterplot for the following data.

Linear Relationships Section 1.1

Express the equation {image} in exponential form

Ch. 12 More about regression

7.5 Exponential and Logarithmic Equations

Logarithmic Functions and Their Graphs

Mrs. Volynskaya Pre-Calculus Exponential & Logarithmic Equations

Exponential & Logarithmic Equations

Graphing Techniques.

Section 5.5 Additional Popper 34: Choice A for #1 – 10

Exponential and Logarithmic Functions

Warmup Solve 256

Transformations to Achieve Linearity

Honors Statistics The Standard Deviation as a Ruler and the Normal Model Chapter 6 Part 3.

MATH 2311 Section 5.5.

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions

Exponential & Logarithmic Equations

Homework: pg. 276 #5, 6 5.) A. The relationship is strong, negative, and curved. The ratios are all Since the ratios are all the same, the exponential.

Exponential & Logarithmic Equations

All pupils can write exponentials as logarithms

Warm-up: Solve for x: CW: Practice Log Quiz HW: QUIZ Review 3.1 – 3.4.

Chapter 8 Section 6 Solving Exponential & Logarithmic Equations

Section 5.5 Additional Popper 34: Choice A for #1 – 10

Section 5.5 Additional Popper 34: Choice A for #1 – 10

MATH 2311 Section 5.5.

Presentation transcript:

1Wednesday, May 10, 20061 Section 4.5 Linearization Process of transforming curvilinear data to make it linear Allows for ease in determining trends and selecting the most appropriate model 4/24/2019 Mr M Kennedy

Model: Consumer Price Index Year Price 1967 100 1968 106 1969 113 1970 120 1971 127 1972 135 1973 143 1974 152 1975 162 1976 172 1977 182 Model: Consumer Price Index The CPI allows comparisons of the costs of goods and services across years. If the base year for $100 of goods and services is 1967, determine the cost of these goods in 2002. Plot the data and determine if it is linear or curvilinear. Let 1967 be year t = 0. 4/24/2019 Mr M Kennedy

CPI Data Scatter Plot and Best Fit Line 4/24/2019 Mr M Kennedy

CPI Data is Curvilinear Is the CPI data modeled by a power function or an exponential function? Linearization of the data allows us to answer this question. 4/24/2019 Mr M Kennedy

Linearizing Exponential Data We use the inverse logarithmic function to linearize exponential data. Suppose our exponential model is 4/24/2019 Mr M Kennedy

Linearizing Exponential Data If the model is exponential, then transforming data (x, y) by taking the Semi-log (x, ln y) will linearize the data. Fit a line to the linearized data. Reverse the above procedure to find the exponential model. 4/24/2019 Mr M Kennedy

Year Price ln(Price) 1967 100 4.605 1968 106 4.663 1969 113 4.727 1970 120 4.787 1971 127 4.844 1972 135 4.905 1973 143 4.963 1974 152 5.024 1975 162 5.088 1976 172 5.147 1977 182 5.204 Linearizing CPI data Natural logarithm of the dependent variable Price results in a nearly constant rate of change of k = 0.06 Indicates the original data has an exponential model 4/24/2019 Mr M Kennedy

Linearization CPI Data: (t, ln p) data and Best Fit Line 4/24/2019 Mr M Kennedy

CPI Exponential Model Best Fit Line for Semi-log data (t, ln p) 4/24/2019 Mr M Kennedy

CPI Exponential Model Determine CPI in 2002 t = 2002 – 1967 so t = 35 4/24/2019 Mr M Kennedy

Model: Territory Weight Area 10 20 50.5 50 168 100 417 150 709 A wild animal maintains a territory which they defend from animals of the same species. How is the weight of the animal related to its territorial area? Not linear, test to see if power function model. 4/24/2019 Mr M Kennedy

Linearizing Power Data We use the inverse logarithmic function to linearize power data. Suppose our power model is 4/24/2019 Mr M Kennedy

Linearizing Power Data If the model is power function, then transforming data (x, y) by taking the Log-log (ln x, ln y) will linearize the data. Fit a line to the linearized data. Reverse the above procedure to find the power model. 4/24/2019 Mr M Kennedy

Linearizing Territory data Natural logarithm of both variables results in a linear trend Indicates the original data has a power model Weight Area Ln(Weight) Ln(Area) 10 20 2.303 2.996 50.5 3.923 50 168 3.912 5.124 100 417 4.605 6.033 150 709 5.010 6.564 4/24/2019 Mr M Kennedy

Territory data Log-log Plot and Line of Best Fit 4/24/2019 Mr M Kennedy

Territory Power Model Best Fit Line for Log-log data (ln w, ln t) 4/24/2019 Mr M Kennedy

Territory Power Model Determine when the territorial area is 800 acres. 4/24/2019 Mr M Kennedy

Graphical Solution: y = w1.32 - 824.74 4/24/2019 Mr M Kennedy

Classroom Participation Find the model for data using linearization 4/24/2019 Mr M Kennedy

Classroom Participation Find the model for data using linearization Use linearization to determine a model for the original data? Is it a power function or an exponential function? 4/24/2019 Mr M Kennedy