Unit 4: Exponential and Logarithmic Functions - Answer Key: A Comprehensive Guide
This guide provides a comprehensive overview of the answers and key concepts for Unit 4, focusing on exponential and logarithmic functions. Because I cannot access specific textbooks or learning materials, this response will offer a general framework for understanding and solving problems related to this unit. You will need to adapt this information to your specific textbook and assignments. Remember to always check your own work against your teacher's key or provided solutions.
I. Exponential Functions:
A. Understanding Exponential Growth and Decay:
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Key Concepts: Exponential functions model situations where a quantity changes by a constant percentage over equal time intervals. Growth occurs when the base of the exponential function (b) is greater than 1 (b > 1), while decay happens when 0 < b < 1. The general form is
y = abˣ, where 'a' is the initial value and 'b' is the growth/decay factor. -
Example Problem: A population of bacteria doubles every hour. If the initial population is 100, what is the population after 3 hours?
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Solution: Here, a = 100 and b = 2 (doubles). The equation is
y = 100 * 2ˣ. After 3 hours (x = 3), y = 100 * 2³ = 800.
B. Graphing Exponential Functions:
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Key Concepts: Exponential graphs show rapid growth or decay. They have a horizontal asymptote (a line the graph approaches but never touches) at y = 0 for typical exponential functions. Understanding transformations (shifts, stretches, and reflections) is crucial for accurate graphing.
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Example Problem: Graph the function
y = 2ˣ + 1. -
Solution: This is an exponential growth function shifted vertically upward by 1 unit. The asymptote is y = 1.
C. Solving Exponential Equations:
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Key Concepts: Solving often involves rewriting equations with the same base, then equating exponents. Logarithms are also essential for solving exponential equations where bases cannot be easily matched.
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Example Problem: Solve
3ˣ = 81. -
Solution: Rewrite 81 as 3⁴. Then, 3ˣ = 3⁴, so x = 4.
II. Logarithmic Functions:
A. Understanding Logarithms:
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Key Concepts: A logarithm is the inverse of an exponential function. The equation
logb(x) = yis equivalent tobʸ = x. Common logarithms (base 10) are written as log(x), and natural logarithms (base e) are written as ln(x). -
Example Problem: Find log₁₀(100).
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Solution: Since 10² = 100, log₁₀(100) = 2.
B. Properties of Logarithms:
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Key Concepts: Understanding the properties of logarithms (product rule, quotient rule, power rule, and change of base) is critical for simplifying and solving logarithmic equations.
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Example Problem: Simplify log(xy²) using logarithmic properties.
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Solution: Using the product and power rules: log(x) + 2log(y).
C. Graphing Logarithmic Functions:
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Key Concepts: Logarithmic graphs are reflections of exponential graphs across the line y = x. They have a vertical asymptote.
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Example Problem: Graph the function
y = log₂(x). -
Solution: This is a logarithmic function with a vertical asymptote at x = 0.
D. Solving Logarithmic Equations:
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Key Concepts: Solving logarithmic equations often involves using the properties of logarithms to simplify the equation, then converting to exponential form.
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Example Problem: Solve log₂(x) = 3.
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Solution: Convert to exponential form: 2³ = x, so x = 8.
III. Applications of Exponential and Logarithmic Functions:
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Key Concepts: These functions model real-world phenomena like compound interest, population growth, radioactive decay, and pH levels.
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Example Problem: If $1000 is invested at 5% annual interest compounded annually, what is the balance after 10 years?
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Solution: Use the compound interest formula: A = P(1 + r/n)^(nt), where A is the balance, P is the principal, r is the rate, n is the number of times interest is compounded per year, and t is the time in years. Here, A = 1000(1 + 0.05/1)^(1*10) ≈ $1628.89.
This is a general framework. Consult your specific unit materials for detailed answers and examples relevant to your coursework. Remember to show your work clearly, and if you are still struggling with specific problems, seek help from your teacher or tutor.
