<p>In this lesson, students will learn to simplify and evaluate exponential and logarithmic expressions. Students will:<br>- comprehend the properties of logarithms and exponents.<br>- evaluate exponential and logarithmic expressions without the use of a calculator.&nbsp;</p>

<p>- How can we decide whether a real-world scenario should be represented by a quadratic, polynomial, or exponential function?&nbsp;<br>- How can you describe the advantages of using multiple approaches to represent exponential functions (tables, graphs, equations, and contextual situations)?&nbsp;<br>&nbsp;</p>

<p>- <strong>Common logarithm:</strong> A logarithm in base 10; if <i>a</i> = \(10^x\), then log <i>a</i> = <i>x</i>.&nbsp;<br>- <strong>Exponential equation:</strong> An equation in the form of <i>y = ax</i>; an equation in which the unknown occurs in an exponent, for example, \(9^{(x+1)}\) = 243.&nbsp;<br>- <strong>Exponential Expression:</strong> An algebraic expression, involving an exponent; if&nbsp;<br>a = \(b^x\), then \(log_{b}a\) = x<br>- Logarithmic equation: An equation in the form of y = \(log_{a}x\), where x = \(a^y\); the inverse of an exponential equation; an equation containing the logarithm of the unknown, for example, log <i>x</i> + 2log 2<i>x</i> + 4 = 0.&nbsp;<br>- Logarithmic Expression: The inverse of an exponential expression.&nbsp;</p>

<p>- a map of the school’s neighborhood&nbsp;<br>- calculators&nbsp;<br>- Matching Worksheet (M-A2-4-1_Matching Worksheet and KEY)&nbsp;<br>- Properties of Exponents and Logarithms Notesheet (M-A2-4-1_Exponents and Logarithms Notesheet and KEY)&nbsp;<br>- Properties Puzzle (M-A2-4-1_Properties Puzzle)&nbsp;<br>- Simplifying-Evaluating Worksheet (M-A2-4-1_Simplifying-Evaluating Worksheet)&nbsp;<br>- Lesson 1 Exit Ticket (M-A2-4-1_Lesson 1 Exit Ticket and KEY)</p>

<p>- In the Think-Pair-Share activity (Part 1), students must relate base and exponent representations to the proper logarithmic expression. Accurate matches show that students understand the link between exponents, bases, and logarithms.&nbsp;<br>- Student performance on the Lesson 1 Exit Ticket will demonstrate how well they understand and portray logarithms. Use student errors to emphasize areas of the lesson that were misunderstood.&nbsp;<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, and Explicit Instructions&nbsp;<br>W: Students will learn to simplify exponential and logarithmic expressions. They are learning how to solve exponential and logarithmic problems. They will be examined via observation, exit tickets, and an assessment.&nbsp;<br>H: After discussing shortcuts, students will be more interested in today's class. Our society values speed and efficiency, therefore students will understand why they are learning today's topic.&nbsp;<br>E: Today, students will work in pairs and independently. They will be able to put together a problem demonstrating the exponential and logarithmic properties.&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 at 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. The properties problem and riddle in the Simplifying and Evaluating Worksheet are intended to tap into some of the skills of kinesthetic learners while also providing additional motivation for all students. There is also an extension problem for students who require additional practice or who perform faster than their colleagues.&nbsp;<br>O: This lesson is divided into four parts, each with either individual or companion work. Go over each problem and debate it as a class. The talks will help the class go from activity to activity.&nbsp;<br>&nbsp;</p>

<p>Start the class by displaying a map of the school's neighborhood with two paths leading "toward any randomly chosen location." One way should be short, while the other should be longer.<br><br><strong>"How many of you would select path A to return home?"</strong> (Students raise their hands.)<br><br><strong>"How many of you would select path B to return home?"</strong> (Students raise their hands.)<br><br><strong>"Why did you pick that path? What elements did you consider?" </strong>(Students who choose path A will presumably say that it is quicker.)<br><br><strong>"We enjoy saving time and getting things done more efficiently. Today, we'll study how to simplify expressions to save time while solving exponential and logarithmic equations. Why do we use exponents?"</strong><br><br>Hopefully, students will remember that exponents are used to summarize repetitive multiplication.<br><br><strong>"Remember that 2 × 2 × 2 is equal to \(2^3\). A logarithm is a sort of exponent. It is the exponent required to raise a number to a different number. Using \(2^3\) = 8, we say base of two raised to the third power. The logarithm is 3 and the base is 2."</strong><br><br><strong>"We can rewrite the formula as follows: 3 = \(log_{2}8\). 2 is the base, 3 is the&nbsp;logarithm of 8. So the formulas \(2^3\) = 8 and 3 = \(log_{2}8\) are equivalent expressions. Remember that when we write a log without a base, it is base ten. This indicates that log 100000 = 5 is equivalent to \(log_{10}100000\) = 5. Not including 10 as the basis is equivalent to writing </strong><i><strong>x</strong></i><strong> and recognizing that it is actually \(x^1\) (</strong><i><strong>x</strong></i><strong> to the first power)."</strong><br><br><strong>"Let's take a more formal look at this. If </strong><i><strong>a = bx</strong></i><strong> is an exponential function, then \(log_{b}a\) = </strong><i><strong>x</strong></i><strong>. Here are some examples." </strong>Write the following on the board:<br><br>If 9 = \(3^2\), then \(log_{3}9\) = 2.<br><br>If 2197 = \(13^3\) , then \(log_{13}2197\) = 3.<br>Similarly, if \(10^{-4}\) =&nbsp; (1/10000), then log 0.0001 = –4.<br><br><strong><u>Part 1</u></strong><br><br><strong>Activity 1&nbsp;</strong><br><br>Distribute the Matching Worksheet (M-A2-4-1_Matching Worksheet with KEY). Students should work on it individually for a few minutes before pairing up. They can utilize their calculators.&nbsp;<br><br>After reviewing the Matching Worksheet with students, have them complete the Properties of Exponents and Logarithms Notesheet (M-A2-4-1_Exponents and Logarithms Notesheet and KEY).<br><br><strong>Activity 2</strong><br><br>After matching up the expressions, students should use the bottom of the worksheet to describe two of their matches. They should explain how they determined that their matches are equivalent. They may struggle with this phase, but allow them some time to consider.<br><br><strong>Activity 3</strong><br><br>Students should work together to complete the Properties Puzzle (M-A2-4-1_Properties Puzzle). Make copies of the page, then cut them into squares.<br><br><strong><u>Part 2</u></strong><br><br><strong>Activity 4</strong><br><br>Once students have completed the puzzle, give them a copy of the Simplifying-Evaluating Worksheet (M-A2-4-1_Simplifying-Evaluating Worksheet). They should be able to finish this worksheet <i>without</i> a calculator.<br><br><strong>Activity 5</strong><br><br>Distribute the Lesson 1 Exit Ticket (M-A2-4-1_Lesson 1 Exit Ticket with KEY) to see whether students comprehend the topics. Students who are having difficulty might visit <a href="http://www.purposegames.com/game/negative-and-zero-exponents-quiz"><span style="color:#1155cc;"><u>http://www.purposegames.com/game/negative-and-zero-exponents-quiz</u></span></a>. They can take a quick quiz to determine how well they grasp negative and zero exponents.<br><br><strong>Extension:</strong><br><br>Students work in pairs, writing an expression and then having their companion write the equivalent or simpler expression.</p>