Week 9 Chapter 5: Exponential and Logarithmic Functions

User Generated

avpxlccunz36

Mathematics

Description

Complete the attached documents and following questions below for exponential and logarithmic functions

  • The bacteria in a 4-liter container double every minute. After 60 minutes the container is full. How long did it take to fill half the container?
  • The graph of a-x and (1/a)x are identical. Why?
  • The graph of logax is increasing for which values of a?
  • Time to think about gathering material to study for the final. Have you? What memory work will you need?

Unformatted Attachment Preview

Graph Exponential Functions; Graph Using Transformations Graph f ( x ) = 2  3x+1 − 4 using transformations and the guided visualization app: https://media.pearsoncmg.com/cmg/pmmg_mml_shared/precalc_ifigs_HTML5/IFig1_29Precalc_graphs_of_exponential_functions/index.html Graph What was the transformation to the graph? What happened? 3𝑥 n/a 3𝑥+1 2 ∙ 3𝑥+1 2 ∙ 3𝑥+1 − 4 Give screenshots of the results for each step. Equation of asymptote Domain Range Section 5.3 & 5.4 Chapter 5: Exponential and Logarithmic Functions Section 5.3: Exponential Functions https://mediaplayer.pearsoncmg.com/assets/kr1HmGZV5rMD_ckX1hN6FQVYkaUIRP_k Author in Action: Evaluate Exponential Functions (17:33) Before delving into exponential functions, let’s make sure we can use our calculators to evaluate exponential expressions. Most calculators have either an x y key or a carot key ^ for working with exponents. To evaluate expressions of the form a x , enter the base a, then press the x y key (or ^ ), enter the exponent x, and press = , (or enter ). Post 1 Example 1*: Evaluating Exponential Functions: Using the Calculator Using a calculator, evaluate: (a) 21.4 (b) 21.41 (c) 21.414 (d) 21.4142 Laws of Exponents If s, t, a, and b are real numbers a  0 and b  0 then, 1) a s  at = _____ 2) (a s )t = _____ 4) 1s = _______ 5) a − s = ____ = ____ (e) 2 3) (ab) s = _____ 6) (a)0 = _____ 2 Exploration 1*: Evaluate Exponential Functions Suppose you are given $10 and told that each day you show up to class the amount you’re given doubles. Fill in the chart below. # of Days you Go to Class 0 1 2 3 4 5 6 7 8 Pay ($) 10 20 Copyright © 2016 Pearson Education, Inc. Exponential and Logarithmic Functions (a) As the value of the independent variable (days) increases by 1, what is happening to the value of the dependent variable (pay)? (b) Create a formula that models this situation. This situation in Exploration 1 models an exponential growth function. Definition: An exponential function is a function of the form ______________ where a is a positive real number (a > 0) and a ≠ 1 and C ≠ 0 is a real number. The domain of f is _________________________. The base a is the ______________ factor, and because f ( 0) = Ca0 = C , C is called the _______________. End of Post 1 Think/Pair/Share: We will discuss this more in depth later –do you have any thoughts about what makes a number a growth factor verses a decay factor? Exploration 2*: Evaluate Exponential Functions: Linear or Exponential? Now that we have studied both linear and exponential functions, we should be able to look at data and determine whether it is either of these functions. But how? Let’s explore this. 1. Evaluate f ( x) = 2x and g ( x) = 3x + 2 at x = −2, −1,0,1, 2, and 3 x f ( x) = 2 x x -2 -1 0 1 2 3 g ( x) = 3x + 2 -2 -1 0 1 2 3 2. Comment on the patterns that exist in the values of f and g. Copyright © 2016 Pearson Education, Inc. Section 5.3 & 5.4 Theorem: For an exponential function, f ( x) = a x , a  1, a  1 , if x is any real number, then f ( x + 1) = _______ or f ( x + 1) = _______ f ( x) Example 2: Identify Linear or Exponential Functions Determine whether the given function is linear, exponential, or neither. For those that are linear, find a linear function that models the data. For those that are exponential, find an exponential function that models the data. (a) (b) . x -1 0 1 2 3 y = f ( x) Average Rate of Change -4.5 -3 -1.5 0 1.5 Ratio of consecutive outputs -1 0 1 2 3 f ( x) = ___________________ (c) x y = g ( x) Average Rate of Change -4.5 -3 -1.5 0 1.5 g ( x) = ___________________ (d) x -1 0 1 2 3 y = h( x) Average Rate of Change 20 16 12 8 4 h( x) = ___________________ Ratio of consecutive outputs Ratio of consecutive outputs . x y = j ( x) Average Rate of Change -1 0 1 2 3 2 3 4.5 6.75 10.125 j ( x) = ___________________ Copyright © 2016 Pearson Education, Inc. Ratio of consecutive outputs Exponential and Logarithmic Functions https://mediaplayer.pearsoncmg.com/assets/jE59a1TlK55_TFoW_sNfnREssnhgnjde Author in Action: Graph Exponential Functions (25:55) Exploration 3*: Graph Exponential Functions 1. Consider the functions f ( x ) = 2x and g ( x ) = 3x . Graph these functions by filling in the table below. Label each of your graphs. x y = f ( x) x -2 -3 -1 -2 0 -1 1 0 2 1 y = g ( x) 2 (a) What are the domain and range of each of these functions? (b) Can y = 0 ? Why or why not? (c) Does the function have any symmetry? (d) What are the x and y – intercepts? (e) Notice that both of these functions are increasing as x increases. What does this mean? Which function is increasing faster? (f) Do these functions have a horizontal asymptote? Copyright © 2016 Pearson Education, Inc. Section 5.3 & 5.4 Post 2 Properties of the Exponential Function f ( x) = a x , a  1 1. The domain is the set of all real numbers or _______________ using interval notation; the range is the set of positive real numbers or __________ using interval notation. 2. There are ____ x – intercepts; the y – intercept is ____. 3. The x – axis (_____) is a horizontal asymptote as _______ [ _______________ ]. f ( x) = a x , a  1 is an ____________ function and is _______________. 5. The graph of f contains the points _______, _______, and _______. 6. The figure of f is smooth and continuous with no corners or gaps. 4. x x 1 1 2. Now consider the functions f ( x ) =   and g ( x ) =   . Graph these functions by filling in the 2 3 table below. Label each of your graphs. x x y = f ( x) y = g ( x) -2 -2 -1 -1 0 0 1 1 2 2 (a) Why do these functions decrease when the graphs in (1) increased (as x gets larger)? (b) What are the domain and range of each of these functions? (c) Do these functions have any asymptotes? Copyright © 2016 Pearson Education, Inc. Exponential and Logarithmic Functions Properties of the Exponential Function f ( x) = a x , 0  a  1 1. The domain is the set of all real numbers or _______________ using interval notation; the range is the set of positive real numbers or __________ using interval notation. 2. There are ____ x – intercepts; the y – intercept is ____. 3. The x – axis (_____) is a horizontal asymptote as _______ [ _______________ ]. f ( x) = a x , 0  a  1 is a ____________ function and is _______________. 5. The graph of f contains the points _______, _______, and _______. 6. The figure of f is smooth and continuous with no corners or gaps. 4. End of Post 2 http://www.mathxl.com/info/exercise.aspx?id=spc10e,5,3,57&se=0 Example 3*: Graph Exponential Functions; Graph Using Transformations x +1 Graph f ( x ) = 2  3 − 4 and determine the domain, range, and horizontal asymptote of f. Make sure you graph and label the asymptote(s). Copyright © 2016 Pearson Education, Inc. Section 5.3 & 5.4 As we saw in our Exploration 1 exponential functions are often used in applications involving money. The act of doubling our money each day says that we are experiencing 100% growth each day. In many examples involving money, we experience growth on a cycle other than per day. Sometimes, our money may grow annually, quarterly, or monthly. These different cycles are called different compounding periods. As we compound more and more often, we say that we are compounding continuously. What is interesting is that as these compound periods approach ∞, we reach a limit. This limit is the number 𝑒. https://mediaplayer.pearsoncmg.com/assets/F8FOfmpR48kG_jqKIpkzGuWg77fKMGg4 Author in Action: Define the Number e (12:39) Exploration 4*: Define the Number e n  1 The number e is defined as e = lim  1 +  . n →  n Let’s explore this value by filling in this table using a graphing utility: n  1 f ( n) =  1 +   n n 10 50 100 500 1000 10,000 100,000 1,000,000 Based on the table, what is e approximately? Confirm the approximate value of e by typing in e into your calculator Copyright © 2016 Pearson Education, Inc. Exponential and Logarithmic Functions ***We will do more applications with the number 𝑒 in Section 6.7*** Example 4*: Define the Number e; Graph e Using Transformations Graph f ( x ) = −e x−2 and determine the domain, range, and horizontal asymptote of f. Solving Exponential Equations Now that we know what exponential functions are let’s learn about how we can solve exponential equations. For example, how would you solve the following: 5 x+ 3 = 1 5 What makes this equation different from equations we’ve seen before? Copyright © 2016 Pearson Education, Inc. Section 5.3 & 5.4 https://mediaplayer.pearsoncmg.com/assets/8neFB_itSvyuIAyU0_WeaHc5t9OdMOLq Author in Action: Solve Exponential Equations (8:16) Post 3 Solve Exponential Equations If a u = a v , then __________ . This means that if you have the same bases on both sides of the equals sign, you set the exponents equal. The key here is to manipulate as needed so that the base is the same. Example 5*: Solve Exponential Equations Solve each equation. (a)* 23 x -1 = 32 (c) 42 x−5 = (e) 5x 2 +8 1 16 = 1252 x (g)* e 2 x −1 = 4 1  e− x ) 3x ( e (b) 5x = 5−6 (d) 22 x−1 = 4 (f) 92 x  27 x = 3−1 2 1 (h)   2 x −5 ( ) = (8x ) 2 x 2 End of Post 3 Copyright © 2016 Pearson Education, Inc. Exponential and Logarithmic Functions We will return to applications for the second discussion post. https://mediaplayer.pearsoncmg.com/assets/elw5aOWOy8AQc3D9Sd_O0TUzOSeTfusL Application of Exponential Function (9:14) Example 6*: Application of Exponential Functions Between 12:00 PM and 1:00 PM, cars arrive at Citibank’s drive – thru at a rate of 6 cars per hour (0.1 car per minute). The following formula from probability can be used to determine the probability that a car will arrive within t minutes of 12:00 PM: F (t ) = 1 − e−0.1t (a) Determine the probability that a car will arrive within 10 minutes of 12:00 PM (that is, before 12:10 PM). (b) Determine the probability that a car will arrive within 40 minutes of 12:00 PM (before 12:40 PM). (c) What values does F approach as t becomes unbounded in the positive direction? (d) Graph F using a graphing utility. (e) Using TRACE, determine how many minutes are needed for the probability to reach 50%. Copyright © 2016 Pearson Education, Inc. Section 5.3 & 5.4 Section 5.4: Logarithmic Functions https://mediaplayer.pearsoncmg.com/assets/rQpLRRVX84aUFwNIsuW2BrgpMwMCz8EU Author in Action: Convert Exponential to Logarithmic Statements & Logarithmic to Exponential Statements (11:35) Post 4 Exploration 1: Logarithms Before we define a logarithm, let’s play around with them a little. See if you can follow the pattern below to be able to fill in the missing pieces to a – f. 1 2 log3 9 = 2 log 9 3 = log 4 16 = 2 log3 27 = 3 (a) log 2 8 = ___ (b) log 4 16 = ___ (c) log ___ 64 = 2 (d) log ___ 64 = 3 (e) log 2 ____ = 4 (f) log 4 2 = ___ Logarithms - A logarithm is just a power For example, log 2 (32) = 5 says “the logarithm with base 2 of 32 is 5.” It means 2 to the 5th power is 32. Notice that both in logarithms and exponents, the same number is called the base. The logarithmic function with base a, where a  0 and a  1 , is denoted by y = log a x (read as “y is the logarithm to the base a of x”) and is defined by: ________________________________ The domain of the logarithmic function y = logax is ___________. Example 1*: Convert Exponential to Logarithmic Statements Change each exponential equation to an equivalent equation involving a logarithm (a) 58 = t (b) x −2 = 12 (c) e x = 10 Copyright © 2016 Pearson Education, Inc. Exponential and Logarithmic Functions Example 2*: Convert Logarithmic to Exponential Statements Change each logarithmic equation to an equivalent equation involving an exponent. (a) y = log 2 21 (b) log z 12 = 6 (c) log 2 10 = a End of Post 4 https://mediaplayer.pearsoncmg.com/assets/mTKoFcOmDv5IpLKcg_3a9nrEAIcEH4MC Author in Action: Evaluate Logarithmic Expressions (4:07) Example 3*: Evaluate Logarithmic Expressions Evaluate the following: 1 (b)* log 2 (a)* log3 (81) (c) log5 (1) 8 (e) log3 (9) (f) log 4 (2) (g) log1/3 (27) (d) log 2 (16) (h) log 5 (25) Copyright © 2016 Pearson Education, Inc. Section 5.3 & 5.4 https://mediaplayer.pearsoncmg.com/assets/_tH8IC95NEzileFMJWwyHmzqxQoecDEz Author in Action: Determine the Domain of a Logarithmic Function (10:01) Let’s recall the domain and range of an exponential function: Range Domain Domain f ( x) = a x All Real Numbers Since a logarithmic function is the inverse of an exponential function, fill in the domain and range below based on what we learned in Section 5.2. Range f ( x) = log a x All Real Numbers greater than 0 Domain and Range of the Logarithmic Function y = log a ( x) (defining equation x = a y ) Domain:__________________ Range:__________________ Example 4*: Determine the Domain of a Logarithmic Function Find the domain of each logarithmic function.  x+3 (a) f ( x ) = log 3 ( x − 2 ) (b) F ( x ) = log 2    x −1  (c) h ( x ) = log 2 x − 1 (d) g ( x ) = log 1 x 2 2 Copyright © 2016 Pearson Education, Inc. Exponential and Logarithmic Functions https://mediaplayer.pearsoncmg.com/assets/GzvFZGCTSZoFcJ2mm_gbAqPqBhZ9tF2r Author in Action: Graph Logarithmic Functions (13:44) Post 5 1. 2. 3. 4. 5. 6. Properties of the Logarithmic Function f ( x) = log a ( x) The domain _______________; The range is _______________. The x-intercept is _______________. There is _______________ y-intercept. The y-axis ( x = 0 ) is a ____________________ asymptote of the graph. A logarithmic function is decreasing if __________ and increasing if __________. The graph of f contains the points ___________________________. The graph is _______________________________, with no _________________. Know natural and common logarithms Fact Natural Logarithm: ln ( x ) means log e ( x ) . It is derived from the Latin phrase, logarithmus naturalis. In other words, y = ln( x) if and only if x = e y . Copyright © 2016 Pearson Education, Inc. Section 5.3 & 5.4 Example 5*: Graph Logarithmic Functions (a)* Graph f ( x) = 3ln( x − 1) . (b)* State the domain of f ( x ) . (c)* From the graph, determine the range and vertical asymptote of f. End of Post 5 (d) Find f −1 , the inverse of f. (e) Use f −1 to confirm the range of f found in part (c). From the domain of f, find the range of f −1 . (f) Graph f −1 on the same set of axis as f. Know natural and common logarithms Fact Common Logarithm: log ( x ) means log10 ( x ) . In other words, y = log( x) if and only if x = 10 y . Copyright © 2016 Pearson Education, Inc. Exponential and Logarithmic Functions Example 6: Graph a Logarithmic Functions (a) Graph f ( x ) = −2log ( x + 2 ) . (b) State the domain of f ( x ) . (c) From the graph, determine the range and vertical asymptote of f. (d) Find f −1 , the inverse of f. (e) Use f −1 to confirm the range of f found in part (c). From the domain of f, find the range of f −1 (f) Graph f −1 on the same set of axis as f. Copyright © 2016 Pearson Education, Inc. Section 5.3 & 5.4 We will return to this next week. https://mediaplayer.pearsoncmg.com/assets/bIl00_lvtNkCqnHF077MHAI55lbv7h_y Author in Action: Solve Logarithmic Equations (6:43) Solving Basic Logarithmic Equations When solving simple logarithmic equations (they will get more complicated in Section 5.6) follow these steps: 1. Isolate the logarithm if possible. 2. Change the logarithm to exponential form and use the strategies learned in Section 5.3 to solve for the unknown variable. Example 7*: Solve Logarithmic Equations Solve the following logarithmic equations (a)* log2 ( 2x +1) = 3 (b)* log x 343 = 3 (c) 6 − log(𝑥) = 3 (d) ln ( x ) = 2 (f) log 6 36 = 5 x + 3 (e) 7 log 6 (4 x) + 5 = −2 Copyright © 2016 Pearson Education, Inc. Exponential and Logarithmic Functions Steps for solving exponential equations of base e or base 10 1. Isolate the exponential part 2. Change the exponent into a logarithm. 3. Use either the “log” key (if log base 10) or the “ln” (if log base e) key to evaluate the variable. Example 8*: Using Logarithms to Solve Exponential Equations Solve each exponential equation. (a) e x = 7 (b)* 2e3 x = 6 (c) e5 x −1 = 9 Copyright © 2016 Pearson Education, Inc.
Purchase answer to see full attachment
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

Explanation & Answer

Attached.

Exponential function outline paper:
The questions are done as follows
1. Filling in the table
2. Plotting the functions for each step
3. Exponential function questions


Running Head: EXPONENTIAL FUNCTIONS

1

Exponential Function
Name
Institution
Instructor

EXPONENTIAL FUNCTIONS

2

Graph Exponential Functions; Graph Using Transformations

Graph

What was the transformation to the
graph?
What ...


Anonymous
Really great stuff, couldn't ask for more.

Studypool
4.7
Trustpilot
4.5
Sitejabber
4.4

Related Tags