<p>In this lesson, students will write and solve exponential and logarithmic equations. Students will:&nbsp;<br>- convert between exponential and logarithmic forms.<br>- use the change of base formulas with the common logarithm and natural logarithm.<br>- solve real-world application problems, using exponential and logarithmic equations.<br>- determine the domain and range of the exponential and logarithmic functions.<br>- Identify the characteristics of exponential and logarithmic function graphs.<br>- convert one representation of an exponential or logarithmic function into another representation.<br>- determine what happens to the graph of an exponential or logarithmic function as the parameters change.</p>

<p>- How are relationships expressed mathematically?<br>- How may data be arranged and portrayed to reveal the link between quantities?<br>- How are expressions, equations, and inequalities utilized to quantify, solve, model, and/or analyze mathematical problems?<br>- How can mathematics help us communicate more effectively?<br>- How may patterns be used to describe mathematical relationships?<br>- How can we utilize probability and data analysis to make predictions?<br>- How may detecting repetition or regularity help you solve problems more efficiently?<br>- How does the type of data effect the display method?<br>- How can mathematics help to measure, compare, depict, and model numbers?<br>- What factors determine whether a tool or method is appropriate for a specific task?<br>- How can we know whether a real-world scenario should be represented as a quadratic, polynomial, or exponential function?<br>- How would you explain the advantages of using multiple approaches to portray polynomial functions (tables, graphs, equations, and contextual situations)?<br>&nbsp;</p>

<p>- <strong>Asymptote:</strong> A line such that a point, tracing a given curve and simultaneously receding to an infinite distance from the origin, approaches indefinitely near to the line; a line such that the perpendicular distance from a moving point on a curve to the line approaches zero as the point moves off an infinite distance from the origin.&nbsp;<br>- <strong>Exponential Equation:</strong> An equation in the form of <i>y</i> = <i>ax</i>; an equation in which the unknown occurs in an exponent, for example, \(9^{(x+1)}\) = 243.<br>- <strong>Logarithmic Equation:</strong> An equation in the form of <i>y</i> = \(\log_{a}x\), where <i>x</i> = \(a^y\); the inverse of an exponential equation.<br>- <strong>Domain:</strong> The set of all <i>x</i>-values or input values for an equation.<br>- <strong>Range:</strong> The set of all <i>y</i>-values or output values for an equation.<br>- <strong>Common Logarithm:</strong> Logarithm with base 10; if <i>a</i> = \(10^x\), then log <i>a</i> = <i>x</i>.<br>- <strong>Natural Logarithm:</strong> Logarithm with base e; also ln, Napierian logarithm, Euler logarithm. The base, <i>e</i>, is approximately 2.71828.</p>

<p>- Solving Exponential and Logarithmic Applications Worksheet (M-A2-4-2_Solving Exponential and Logarithmic Applications Worksheet)&nbsp;<br>- Lesson 2 Exit Ticket (M-A2-4-2_ Lesson 2 Exit Ticket)&nbsp;<br>- Graphing Exponential and Logarithmic Function Notes (M-A2-4-2_Graphing Exponential and Logarithmic Function Notes and KEY)&nbsp;<br>- Graphing Practice Worksheet (M-A2-4-2_Graphing Practice Worksheet)&nbsp;<br>- graph paper</p>

<p>- The Think-Pair-Share activity (Part 2) utilizes the Graphing Practice Worksheet. Students can assess their own and their partners' comprehension of how to represent logarithmic functions graphically. Remind students to use the ideas they already understand about functions to ensure that their graphs contain only one y-value for each x.&nbsp;<br>- The Lesson 2 Exit Ticket includes a growth/decay model of a real-world logarithmic application and needs students to learn how to apply logarithms to depict a practical situation. Before completing the activity, ask students to think about how rational their answers are.</p>

<p>Scaffolding, Active Engagement.&nbsp;<br>W: Students will learn to solve exponential and logarithmic equations. Solving equations is a key step in generating predictions about various scenarios. Students will be evaluated using observation, exit tickets, and an assessment. Students will also learn how to graph exponential and logarithmic functions. Graphs are significant visual representations of functions. Students will be evaluated using observation, exit tickets, and an assessment.&nbsp;<br>H: Students at this age enjoy crime dramas and mysteries, making today's lesson appealing to them. Many real-world scenarios use exponential and logarithmic functions.&nbsp;<br>E: Today, students will work in pairs and independently. They will make notes that they will use to fulfill the lesson's tasks.&nbsp;<br>R: During the class review, students can reflect on and revisit their concerns. Students will then use that information to revise their mental processes for the next task. You will wander around while the students are working and provide feedback during this period.&nbsp;<br>E: Students can evaluate themselves by reviewing their work with a companion. Their peers may be able to provide more insight into their comprehension.&nbsp;<br>T: This class focuses on teamwork by grouping students based on their ability levels. There is also an extension problem for students who require additional practice or who perform faster than their colleagues.&nbsp;<br>O:&nbsp;This lesson is divided into sections, each of which includes either independent or partner activities. We will review each problem and debate it as a class. The talks will help the class go from activity to activity.&nbsp;<br>&nbsp;</p>

<p><strong>This lesson can be enjoyable for students since it demonstrates how exponential and logarithmic functions are applied in the real world. Ask students if they enjoy crime shows or solving mysteries. Have a discussion about these shows or mysteries, and what students enjoy most about them.</strong><br><br><strong>"Today, we will learn how exponential and logarithmic equations are utilized to solve real-world problems. Who can tell me what the most fundamental exponential equation is and what each component of the equation represents?"</strong>&nbsp;[<i>y</i> = <i>a</i>\(b^x\) or <i>y</i> = <i>a</i>\(b^x\) + <i>k</i>; <i>a</i> ≠ 0 (starting value); <i>b</i> is greater than 0 and ≠ 1 (multiplier: represents a percentage increase or decrease); and <i>k</i> = asymptote (a number that the function approaches but never reaches).]<br><br><strong><u>Part 1</u></strong><br><br>Exponential and logarithmic functions are used in the real world. Exponential functions are most commonly used to describe population growth, interest, and bacterial growth. Logarithmic functions are used to measure the intensity of light and sound, as well as the magnitude of earthquakes. Since students will be graphing logarithmic equations, go over how to convert between exponential and logarithmic forms.<br><br><strong>"Today we will learn how to graph exponential and logarithmic functions without the need for a calculator. We'll start with the equation, create a table of values with a few points, and then draw the graph."</strong><br><br><strong>"Suppose we have the exponential function </strong><i><strong>y</strong></i><strong> = 3</strong><i><strong>x</strong></i><strong>, we can graph the function using a table of numbers."</strong><br><br><strong>"Let's fill in our table." </strong>Use a projector or interactive whiteboard to show the following chart:&nbsp;</p><figure class="image"><img style="aspect-ratio:367/522;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_67.png" width="367" height="522"></figure><p><strong>"Now, we can create our graph." </strong>Show the following graph:</p><figure class="image"><img style="aspect-ratio:492/364;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_68.png" width="492" height="364"></figure><p><strong>"Take note that </strong><i><strong>y</strong></i><strong> = 0 is approaching on the graph's horizontal asymptote."</strong><br><br><strong>"We're going to graph a logarithmic function now!"</strong><br><br><strong>"Because a logarithmic function is the inverse of an exponential function, we simply graph the inverse, draw the line of symmetry, y = x, and plot the reverse coordinates for each point on the exponential function. An illustration will help you understand this process."</strong><br><br><strong>"Let's use our previously defined exponential function, </strong><i><strong>y</strong></i><strong> = \(3^x\). The inverse of this function is log3</strong><i><strong>x</strong></i><strong>.</strong><br><br><strong>"We should revisit our previous table and add an additional column for the ordered pair that contained the reversed coordinates."&nbsp;</strong></p><figure class="image"><img style="aspect-ratio:426/408;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_69.png" width="426" height="408"></figure><p><strong>"We are going to plot the logarithmic function's points now." </strong>Show the following graph:</p><figure class="image"><img style="aspect-ratio:508/382;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_70.png" width="508" height="382"></figure><p><strong>"All we have to do now is link the points of the logarithmic function. Consider the vertical asymptote of </strong><i><strong>x</strong></i><strong> = 0."</strong><br><br><strong><u>Part 2</u></strong><br><br><strong>"Before we can get into the application problems, we need to understand a few formulas. Let's imagine we need to solve 5</strong><i><strong>x</strong></i><strong> = 50. What should we do to solve for </strong><i><strong>x</strong></i><strong>?"</strong> (<i>Divide both sides by five.</i>) <strong>"Dividing is the inverse of multiplying. Thus, what is the inverse of exponents?</strong> (<i>logarithms</i>)<br><br><strong>"We can easily rewrite 8 as \(2^3\), set the exponents equal to each other, and solve for </strong><i><strong>x</strong></i><strong> if we have a problem like \(2^{x-1}\) = 8."</strong><br><br>\(2^{x-1}\) = 8<br><br>\(2^{x-1}\) = \(2^3\)<br><br><i>x</i> − 1 = 3<br><br><i>x</i> = 4<br><br><strong>"But what if we don't have the same base to work from? We can calculate the logarithm of both sides of the equation." </strong>Put the following formulas on the board and practice the examples as a class.<br><br><strong>\(\log{}b^x\) = log </strong><i><strong>a</strong></i><br><strong>or</strong><br><strong>x log </strong><i><strong>b</strong></i><strong> = log </strong><i><strong>a</strong></i><br><br><strong>"Remember that ln represents the natural logarithm, not the base 10 logarithm. It's important to remember that these are two different bases. The natural logarithm has a base of roughly 2.71828 and is widely used in mathematics."</strong><br><br><strong>"The Common Log, base 10, is written when there is no base next to it."&nbsp;</strong><br><br><strong>"Let's try some examples." The examples should be developed as a class. Take note that these equations can be solved in a variety of ways</strong><br><br><strong>1. \(2^x\) = 10</strong><br><strong>Answer for number 1:&nbsp;</strong><br><br><strong>x = log10log2</strong><br><br><strong>2. \(2^x\) = 10</strong><br><strong>Work for number 2:</strong><br><br><strong>log\(2^x\) = log10</strong><br><br><strong>xlog2 = log10</strong><br><br><strong>x = log10log2</strong><br><br><strong>3. \(5^x\) = 45</strong><br><strong>Work for number 3:</strong><br><br><strong>log\(5^x\) = log45</strong><br><br><strong>xlog5 = log45</strong><br><br><strong>x = log45log5&nbsp;</strong><br><br><strong>4. \(8^{x-1}\) = 100</strong><br><strong>Work for number 4:</strong><br><br><strong>log8x - 1 = log100</strong><br><br><strong>(x - 1)log8 = log100</strong><br><br><strong>x - 1 = log100log8</strong><br><br><strong>x = log100log8 + 1</strong><br><br><strong>5. \(6^{2x+3}\) = 50</strong><br><strong>Work for number 5:</strong><br><br><strong>log\(6^{2x+3}\) = log50</strong><br><br><strong>2x + 3log6 = log50</strong><br><br><strong>2x + 3 = log50log6</strong><br><br><strong>2x = log50log6 - 3</strong><br><br><strong>x = (log50log6 - 3) / 2</strong><br><br><strong>"Consider a natural logarithm example. Assume we have the exponential equation: 4</strong><i><strong>e</strong></i><strong>3</strong><i><strong>x</strong></i><strong> + 5 = 10. We can answer the equation using the natural logarithm because </strong><i><strong>e</strong></i><strong> is a base. The base of the natural logarithm, </strong><i><strong>e</strong></i><strong>, functions similarly to base 10. The logarithm is the inverse of the exponential function. Log (1000) = 3 because \(10^3\) equals 1000. Similarly, \(e^3\) ≈ 20.08553, implying that ln (20.08553) ≈ 3." </strong>Work through this with the students:&nbsp;<br><br>4<i>e</i>3<i>x </i>+ 5 = 10<br><br>4<i>e</i>3<i>x</i> = 5<br><br><i>e</i>3<i>x</i>=1.25<br><br>ln<i>e</i>3<i>x</i> = ln1.25<br><br>3<i>x</i> = ln1.25<br><br><i>x</i> = ln1.253<br><br><i>x</i> ≈ .07<br><br><strong>"If we are given a logarithm to evaluate, we can utilize the change of base formula. We can also convert logarithms to different bases."</strong><br><br><strong>"Let's first examine how to use the change of base formula to evaluate a logarithm in terms of common logarithms."</strong> (Review this concept with the whole class.)<br><br>\(\log_{b}M\) = \(\log_{c}M\)\(\log_{c}b\)<br><br>*For all positive numbers <i>b</i>, <i>c</i>, and <i>M</i>, where <i>b</i> ≠ 1 and <i>c</i> ≠ 1.<br><br><strong>"For example, take this logarithm."</strong><br><br>\(\log_{2}14\) = \(\log_{}14\)\(\log_{}2\)<br><br>≈ 3.81<br><br><strong>"Now we are able to convert this logarithm to a different base. Let's convert it to base 6." </strong>Work through this with the students:<br><br>\(\log_{2}14\) = log6<i>x</i><br><br>3.81 ≈ log6<i>x</i><br><br><i>x</i> ≈ 63.81<br><br><i>x</i> ≈ 922.05<br><br><strong>"Now we'll get into some fun problems. We'll do an example as a class, and then you'll work in groups to solve some problems."</strong><br><br>Give the following problem:<br><br>Aunt Helen enjoys drinking tea, but she is specific about the temperature at which she drinks it. She heated the water (100°) and poured it over the tea leaves. Five minutes later, she returned, and the tea had reached 65°. Aunt Helen keeps her house at a cool 20°. Write an equation to reflect the temperature of Aunt Helen's tea.<br><br><strong>"First, we must determine what we know. We know that when </strong><i><strong>t</strong></i><strong> = 0 (the time at which the water comes to a boil), </strong><i><strong>y</strong></i><strong> (the temperature of the tea) is 100, and when </strong><i><strong>t</strong></i><strong> = 5 (the number of minutes after it comes to a boil), </strong><i><strong>y</strong></i><strong> is 65. What more do we know about the problem?"</strong> (Room temperature is 20° Celsius, which is the asymptote, as nothing can cool down faster than room temperature.) <strong>"We'll start by substituting the first point into an exponential equation and solving for </strong><i><strong>a</strong></i><strong>. Then, we'll insert the second point and solve for </strong><i><strong>b</strong></i><strong>. This will give us the percent rate at which the tea cools per minute."</strong> Work through this with the students:<br><br>100 = <i>a</i>\(b^0\) + 20<br>80 = <i>a</i>(1)<br><i>a</i> = 80<br><br>65 = 80\(b^5\) + 20<br>45 = 80\(b^5\)<br>0.5625 = \(b^5\)<br><i>b</i> ≈ 0.8913<br><br>y = 80\((.8913)^t\) + 20<br><br><strong>"We can use this equation to forecast. Let's say Aunt Helen only drinks her tea at 50°. How long will she have to wait to drink her tea?"</strong><br><br>50 = 80\((0.8913)^t\) + 20<br>30 = 80\((0.8913)^t\)<br>0.375 = \((0.8913)^t\)<br><br><strong>"To solve for </strong><i><strong>t</strong></i><strong>, we must now apply the change of base formula."</strong><br><br><i>t</i> = log 0.375 ÷ log 0.8913 or ln 0.375 ÷ ln 0.8913 ≈ 8.5 minutes<br><br><strong>Activity 1</strong><br><br>Hand out the Worksheet for Solving Exponential and Logarithmic Applications (M-A2-4-2). Students should complete this assignment in groups because answering application difficulties can be challenging for some. Once everyone has completed the worksheet, go over it. Students may want to wait to answer the graph question until they have completed both portions of the lesson.<br><br><strong>Activity 2</strong><br><br>Distribute the Graphing Exponential and Logarithmic Functions Notes (M-A2-4-2_Graphing Exponential and Logarithmic Function Notes and KEY). Review the notes and practice problems with the students.<br><br><strong>Activity 3</strong><br><br>Hand out the Graphing Practice Worksheet (M-A2-4-2_Graphing Practice Worksheet). Have students start this worksheet on their own. After a while, students can team up with a partner to double-check their work. When everyone is finished, have students present their answers and work to the board.<br><br><strong>Activity 4</strong><br><br>Distribute the Lesson 2 Exit Ticket (M-A2-4-2_ Lesson 2 Exit Ticket) to see whether students comprehend the concepts.<br><br><strong>Extension:</strong><br><br>1. Using the Graphing Practice Worksheet, consider what would happen if the parameters were adjusted to a different number. For instance, you can ask students the following questions:<br><br>2.<strong> "What if </strong><i><strong>k</strong></i><strong> were −5 in #1? What would we have to do with our present graph?" </strong><i>(shift the graph eight units downward)</i><br><br>3.<strong> "What if </strong><i><strong>h</strong></i><strong> were 3 in #2? What would we have to do with our present graph?"</strong> <i>(shift the graph four units to the right)</i><br><br>4. Ask students to write their own equations and have a partner graph them.</p>