Chapter 4 Exponential and Logarithmic Functions 15 Days

Chapter 4 - Exponential and Logarithmic Functions 15 Days

4. 1 Exponential Functions One Day

What is an exponential function? � Exponential � An functions are those that have example of an exponential function is

Definition of Exponential an Function � An Exponential Function f with base a

Family of Exponential Functions

Sketching Exponential Graphs � Sketch Graphs for the following function:

Sketching Exponential Graphs � Sketch Graphs for the following function:

Shifting Exponential Graphs � We can shift exponential function using the same patterns from before. Use locator point (0, 1).

Reflections of Exponentials - Both types of reflections will change the position(s) of your intercepts and should be done before shifting.

Theorem: Exponential Functions are 1 to 1.

Solving Exponential Equations � To solve equations with variables in the exponents we need to: � 1. Re-write both sides as the same base using exponent rules. � 2. Set the exponents equal using condition 2 of our theorem on exponential functions. � 3. Solve for the variable.

Solve the following exponential equations:

Writing Equations of Exponentials Given Initial Conditions � Writing an exponential function the y-int and a point on the function. � 1. given Substitute the y-int into your equation and solve for b. � 2. Re-write your equation with a value for b. � 3. Substitute other point into your equation from step 2 and solve for a. � 4. Re-write your equation with values for a and b.

Writing Equations of Exponentials � Find an exponential function of the form that has the given y-int and passes through the point P

Homework � Read 4. 1; pg 292 (# 2, 4, 7, 9, 11 b - h, 13, 14, 16 - 18, 25 NO TI’s for the graphs)

4. 2 The Natural Function Two Days

Compound Interest

Compound Interest � You have an account that returns 7% annual interest compounded monthly. If you invest $1500 for a total of 10 years, how much money will you have in the account?

The Number e

The Natural Exponential Function

Continuously Compounded Interest Formula

Continuously Compounded Interest � An initial investment of $35000 is continuously compounded at 8. 5% interest. How much is the investment worth after 5 years? After 15 years?

Law of Growth and Decay Formula

Modeling Population Growth � Since 1980, world population in millions closely fits the exponential function defined by where x is the number of years since 1980. � The world populations was about 5, 320 million in 1990. How closely does the function approximate this value? � Use this model to approximate the population in 2012.

Homework � Read 4. 2; pg 303 (# 1 -7, 9, 11 -13, 15, 19, 21, 25)

Homework � pg 292 (# 32, 37, 38, (53 use TI)); � pg 304 (# 22, 24 (45 & 51 use TI))

4. 3 Logarithms Four Days

Converting to and From Logarithms The following expressions are equivalent. Examples 1) 32=9 3) 2) xa+b=9 4)

Things to note � The expression above is read “The log of x base a equals y” � x>0 (the number you take a log of must be positive) � If you don’t see an “a” value the base is assumed to be 10. ◦ log 4 = log 10 4 �A natural log (ln) has a base of e. ◦ ln 4=loge 4

Log Equations 1) 2) 3) 4)

Homework 4. 3. 1 � Pg 317 #1, 3, 9, 11, 14(skip e), 17 -27 odd. � No TI

� Begin 4. 3. 2

Shortcuts with logs Evaluate 1) 2) 3)

Graphing logs � Remember that f(x)=logax and f(x)=ax are inverse functions. � This means that logarithmic functions will look like exponential functions except their x’s and y’s will be flipped. a) b) Graph f(x)=log 2 x Graph f(x)=log 2(x+2)-1

Domain and range of logarithmic functions �Domain of f(x)=log x �Range of f(x)=log x �Remember (0, ∞) (-∞, ∞) you can’t take the log of a negative number or zero. �However… logs can equal negative numbers. Ex: log(1/2)

Homework 4. 3. 2 � Pg 317 #4, 10, 12, 16, 33(a-g) No calc � 47, 59, 63, 65

� Begin 4. 3. 3

Graphing More Complicated Logs ◦ Idea: Rewrite in exponential form. Plug in y’s to find x’s � Graph f(x)=log 4(2 x-1) � Find asymptotes, intercepts, domain and range

Homework 4. 3. 3 � Logarithmic Functions Worksheet � pg 59 # 8, 11 - 14 � pg 60 1 -3, 5 - 17, 20, 21 � graph 20 & 21(no TI)

4. 4 Properties of Logarithms One Days

Properties of Logs � Loga(xy)= Logax+Logay � Loga(x/y)= Logax-Logay � Logaxn= n. Logax � Note: log(x+y) is not equal to log (x)+ log(y) � Note: log(x-y) is not equal to log (x)-log(y)

Properties of Logs � Loga(xy)= Logax+Logay � Loga(x/y)= Logax-Logay � Logaxn= n. Logax � 1) Expand Each Log 3(4 x) Write as a single log 4) 3 log(x)+2 log(y) 2) Ln(3 e) 5) ½Ln(4 x)-y. Ln(6) 3) Log(2 x 3/y 4) 6) 2 ln(xy)-3 ln(x)+6 ln(y)

Solving Equations � Solve 1) Log 4(2 x+4)= 2 log 43+ log 45 2) ln(x)+ln(x+3)=½ln(324)

Homework 4. 4. 1 � Pg 328 #1 -15 odd, 18, 20, 22 -26

4. 5 Solving Exponential and Logarithmic Equations Three Days

Solving Exponential and Logarithmic Equations

Change of Base Formula

Solving Exponential and Logarithmic Equations

Solving Exponential and Logarithmic Equations

Solving Exponential and Logarithmic Equations

Solving Exponential and Logarithmic Equations

Solving Exponential and Logarithmic Equations

Homework � Read 4. 5 � pg 339 (# 1 -3, 5, 9, 10, 17, 18, 20, 41, 42, 45)

Homework � pg 340 (# 11, 13, 15, 21, 22, 25, 31, 32, 43, 44, 57)

Applications � How long does it take for an initial investment of $5000 to grow to $60000 in an account that earns 8. 5% interest compounded monthly?

Applications populations N(t) (in millions) of the United States t years after 1980 may be approximated by the formula. � When will the populations be twice what is was in 1980? � The

Applications �A 100 g sample of a radioactive substance has a half life of 30 minutes. After how many hours will 20 g remain?

Homework � Solving Equations Worksheet � Review for Quiz