CC 8 Logs, Exponentials, Logistic Growth and Inverses
CC 8 Logs, Exponentials, Logistic Growth and Inverses.
Dec 29, 2015
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CC 8 Logs, Exponentials, Logistic Growth and Inverses
What question comes to mind?
How tall will Chloe be as an adult?
Date Height (inches)
7/16/2010 21.25
9/13/2010 23.75
1/17/2011 27.13
4/18/2011 28.25
7/18/2011 30.5
1/16/2012 34
7/17/2012 36.25
7/22/2013 39.25
7/30/2014 42.5
padlet.com/cyndia_acker/CC8
I. Logistic Growth
A. Definition
Logistic Growth models real-life quantities whose growth levels off because the rate of growth changes from an increasing growth rate to a decreasing growth rate.
On your desk sketch a graph that changes
from increasing to decreasing
B. Visual
Why does the graph
continue to increase if the growth
rate is decreasing?
C. Process
Analyze the function
C. Process
Goal Problems
Recall & Reproduction
If, at the end of two years a savings account has a balance of $1172.60, and the interest rate is compounded monthly at 3.2%, then what is the original amount deposited two years ago?
Routine
Based on recent census data, a logistic model for the population of Dallas, t years after 1900 is as follows.
According to this model, when was the population 1 million?
Non-Routine
Goal Problems
Recall & Reproduction Routine
1985
Non-Routine
Goal Problems
R& R What are the key characteristics of log, exponential and logistic
growth functions? When is compound interest used vs. continuous compound
interest? How do you know when the model is growth or decay?
Routine Why is the y value of the point of maximum growth ? Why do we start out with t = 0?
Non-routine In what context could a domain of all real numbers make sense?
Active Practice
R& R: Analyzing graphs of logs, exponentials and logistic
growth Compound Interest Exponential Growth
Routine: Application of logs, exponentials and logistic growth
Non-Routine Update Plan: How tall will Chloe be as an adult
Task
Exponential & Logarithmic Fcns
LT 3G: I understand the inverse relationship between exponential and logarithmic functions and can use this relationship to explain the definition of a logarithm. I can use the definition of a logarithm to find the value of a variety of given logarithms, including natural logarithms. I can use the inverse relationship between exponential and logarithmic functions to solve algebraic equations or to sketch logarithmic and exponential functions. I can explain when technology is a tool or a hindrance in computing logarithms (e.g. ln1, lne2)
A. Definition
Logarithm – a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.
B. Visual
Log636=2
62=36
What is the general form for the transformation above?
y = bxx = logby
Rewrite the following equation to a more familiar form.
Based on the definition, label the corresponding parts
between the two forms.
.
A. Definition
Logarithm – a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.
y = bxx = logby
C. Process
3216log6 236log6
Given
1216
36log6
Justify why the following equation is a true statement:
Write the general form of this relationship as a logarithm and an exponential.
Rules of Logarithms Rules of Exponents
x = logby
logb(mn) = logbm + logbn
logb(m÷n) = logbm – logbn
logb(mn) = nlogb(m)
Change-of-base formula
C. Process
Notation: Common Logarithm: log x =
log10xNatural Logarithm: ln w = logew
Rules of Logarithms Rules of Exponents
x = logby y = bx
logb(mn) = logbm + logbn
(am)(an) = am+n
logb(m÷n) = logbm – logbn
(am)÷(an) = am-n
logb(mn) = nlogb(m) (am)n = amn
Change-of-base formula
Leading into the Weekend
Reminder: Quick Check Monday (2/23) Topic: Unit 3 CC 7 – Matrices
Agenda
Active Practice CC 7 - Matrices CC 8 - Logs/Exp
Concept Check
Concept Check (Individually)On your own, write down the answers to these
questions:
Explain what happens when you multiply a matrix with its inverse?
Compare and contrast “logistic” vs “logarithmic”?
How are the log rules related to the exponent rules?
Prove that: logb(jk) = logbj + logbk
WAIT! Don’t share thoughts… we will do that after everyone has had time to process on their own.
Concept Check (as a Group)
As your group shares:
In your notes: Write down questions and answers that your group shared that helped you. Tally for yourself, how many times someone writes down one of your questions/answers.
Explain what happens when you multiply a matrix with its inverse?
Compare and contrast “logistic” vs “logarithmic”?
How are the log rules related to the exponent rules?
Prove that: logb(jk) = logbj + logbk
Leading into the Weekend
Reminder: Quick Check Monday (2/23) Topic: Unit 3 CC 7 – Matrices
Make a Plan to prepare for the QC
List either: Questions regarding material that need to
be answer this weekend Problems types that need to be reviewed
Goal: Finding insights to add to your notes
Goal ProblemsRecall & Reproduction:
1. Simplify:
a. log4(16)
b. 6log(100)*e0.
c. log(1/2)32
d. (log100)÷(log10)
e. What is the value of n that satisfies logn64 = 2
Routine:Solve: ln(x + 5) = ln(x – 1) – ln(x + 1)
Non-Routine:The half-life of radioactive radium (226Ra) is 1556 years. What percent of a present amount of radioactive radium will remain after 100 years?
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