Grade 12 · Mathematics
Quarter 4 · Week 4 · TEKS
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This lesson focuses on helping Grade 12 students use the properties of logarithms to evaluate and transform logarithmic expressions. Students will also generate and solve logarithmic and exponential equations in both mathematical and real-world contexts. The lesson aligns with TEKS standards for Precalculus and emphasizes algebraic reasoning through process standards in mathematics.
Students should understand the definition of logarithms and exponents, the relationship between logarithms and exponents, and basic algebraic manipulation skills.
A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, \( \log_b a = c \) means \( b^c = a \). Understanding this relationship is fundamental to manipulating logarithmic expressions.
There are three main properties used to evaluate or transform logarithmic expressions: 1. Product Property: \( \log_b (MN) = \log_b M + \log_b N \). This means the logarithm of a product is the sum of the logarithms. 2. Quotient Property: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \). The logarithm of a quotient is the difference of the logarithms. 3. Power Property: \( \log_b (M^p) = p \log_b M \). The logarithm of a power is the exponent times the logarithm of the base.
To simplify logarithmic expressions, identify parts of the expression that match the product, quotient, or power forms. For example, \( \log_2 (8 \times 4) \) can be rewritten using the product property as \( \log_2 8 + \log_2 4 \). Then evaluate each logarithm separately.
To solve logarithmic equations, use properties to combine or expand logarithms to isolate the variable. For example, if \( \log_b x + \log_b (x-3) = 1 \), use the product property to write \( \log_b [x(x-3)] = 1 \), then rewrite in exponential form \( b^1 = x(x-3) \) and solve the resulting quadratic equation.
When the variable is in the exponent, take the logarithm of both sides to solve. For example, to solve \( 3^x = 81 \), take \( \log_3 \) of both sides: \( x = \log_3 81 \). Since \( 81 = 3^4 \), \( x = 4 \). For bases that are not easily related, use the change of base formula or common logarithms.
Logarithms are used in many real-world contexts such as measuring sound intensity (decibels), pH in chemistry, and exponential growth or decay models. Present problems where students must set up and solve logarithmic or exponential equations based on given scenarios.
Simplify the expression \( \log_3 27 + \log_3 9 \).
Using the product property of logarithms, \( \log_b M + \log_b N = \log_b (MN) \). So, \( \log_3 27 + \log_3 9 = \log_3 (27 \times 9) = \log_3 243 \). Since \( 3^5 = 243 \), \( \log_3 243 = 5 \).
5
Solve the logarithmic equation \( \log_2 (x) + \log_2 (x-2) = 3 \).
Use the product property: \( \log_2 (x) + \log_2 (x-2) = \log_2 [x(x-2)] = 3 \). Rewrite in exponential form: \( 2^3 = x(x-2) \) which is \( 8 = x^2 - 2x \). Rearrange: \( x^2 - 2x - 8 = 0 \). Factor: \( (x-4)(x+2) = 0 \). Solutions: \( x=4 \) or \( x=-2 \). Since logarithms require positive arguments, \( x=4 \) is valid.
x = 4
Evaluate \( \log_5 125 \) using the power property of logarithms.
Since \( 125 = 5^3 \), \( \log_5 125 = \log_5 (5^3) \). Using the power property, \( \log_b (b^p) = p \), so \( \log_5 125 = 3 \).
3
Solve the exponential equation \( 4^x = 64 \).
Rewrite 64 as a power of 4: \( 64 = 4^3 \). Since the bases are equal, \( x = 3 \).
x = 3
Simplify \( \log_2 \left( \frac{16}{4} \right) \) using logarithmic properties.
Use the quotient property: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). So, \( \log_2 \left( \frac{16}{4} \right) = \log_2 16 - \log_2 4 \). Since \( 16 = 2^4 \) and \( 4 = 2^2 \), this is \( 4 - 2 = 2 \).
2
Introduce the lesson by asking students if they have seen logarithms before and what they know about them. Write a simple logarithmic expression on the board and ask students to share what they think it means.
Students share prior knowledge and initial thoughts about logarithms and their properties.
This phase activates prior knowledge and sets the stage for learning about properties of logarithms.
Provide students with several logarithmic expressions and ask them to use calculators or reasoning to evaluate or simplify them. Encourage students to look for patterns or relationships.
Students work in pairs or small groups to evaluate logarithmic expressions and discuss patterns they notice.
Students explore logarithmic expressions to discover how properties can help simplify or evaluate them.
Explicitly teach the three main properties of logarithms with examples. Demonstrate how to apply each property to simplify expressions and solve equations.
Students take notes and ask questions. They practice applying each property to example problems provided by the teacher.
This phase provides direct instruction on the properties of logarithms and their use in evaluating and transforming expressions.
Present real-world problems that can be modeled with logarithmic or exponential equations. Guide students to set up and solve these equations using logarithmic properties.
Students work individually or in groups to translate real-world problems into logarithmic or exponential equations and solve them.
Students apply their understanding of logarithmic properties to solve practical problems, deepening their conceptual understanding.
Administer an exit ticket with problems requiring simplification of logarithmic expressions and solving logarithmic or exponential equations.
Students complete the exit ticket individually to demonstrate their understanding.
This phase assesses student mastery of the lesson objectives.
Students work in pairs or small groups to evaluate logarithmic expressions and discuss patterns they notice.
Simplify \( \log_3 27 + \log_3 9 \).
5
Solve \( \log_2 (x) + \log_2 (x-2) = 3 \).
x = 4
Evaluate \( \log_5 125 \).
3
Solve \( 4^x = 64 \).
x = 3
Simplify \( \log_2 \left( \frac{16}{4} \right) \).
2
Provide step-by-step guided examples with scaffolding. Use visual aids to show how logarithmic properties work. Allow use of calculators for evaluation tasks.
Use clear, simple language and define key vocabulary explicitly. Use visual models and diagrams to illustrate logarithmic properties. Pair ELL students with peers for collaborative learning.
Challenge students with problems involving change of base formula. Introduce solving logarithmic equations with different bases. Explore applications of logarithms in more complex real-world contexts.
Ask students to find examples of logarithms in real life, such as in music, science, or technology, and explain how logarithms are used in those contexts.
In this lesson, students learned the key properties of logarithms—the product, quotient, and power properties—and how to use them to simplify logarithmic expressions. They practiced solving logarithmic and exponential equations by applying these properties and rewriting equations in exponential or logarithmic form. Students also connected these skills to real-world problems, strengthening their algebraic reasoning and problem-solving abilities in line with TEKS standards for Precalculus.
Use the exact standards alignment and retrieved-source provenance stored with this enrichment.