unit 4 exponential and logarithmic functions answer key

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31-01-2025

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Unit 4: Exponential and Logarithmic Functions - Answer Key: A Comprehensive Guide

This guide provides comprehensive answers and explanations for common problems encountered in Unit 4: Exponential and Logarithmic Functions. This unit often covers a broad range of topics, so this key will address several key areas. Remember to always refer to your specific textbook and lesson materials for the most accurate and relevant answers to your assignments.

Note: Because I cannot access your specific textbook or assignment questions, this answer key will focus on common problem types within this unit. You will need to adapt these examples to your particular problems.

Section 1: Exponential Functions

Understanding Exponential Growth and Decay:

• Identifying the Base: The base of an exponential function (e.g., f(x) = abˣ) determines whether the function represents growth (b > 1) or decay (0 < b < 1). The larger the base (for growth), the faster the growth rate. For decay, a base closer to 0 indicates faster decay.

• Solving Exponential Equations: Many problems involve solving for 'x' in equations like 2ˣ = 16. This often requires changing the equation to have the same base on both sides (e.g., rewriting 16 as 2⁴) or using logarithms.

Example: Solve for x in 3ˣ = 81.

Solution: Rewrite 81 as 3⁴. Therefore, x = 4.

• Modeling Real-World Problems: Exponential functions model numerous real-world scenarios like population growth, compound interest, and radioactive decay. Understanding the context is crucial for correctly interpreting results.

Example: A population grows according to the formula P(t) = 1000(1.05)ᵗ, where t is in years. Find the population after 5 years.

Solution: Substitute t = 5 into the formula: P(5) = 1000(1.05)⁵ ≈ 1276.

Section 2: Logarithmic Functions

Understanding Logarithms:

• Definition of a Logarithm: A logarithm is the inverse of an exponential function. The equation logₐb = c is equivalent to aᶜ = b. Understanding this relationship is fundamental to solving logarithmic equations.

• Properties of Logarithms: Mastering the properties of logarithms—product rule, quotient rule, and power rule—is critical for simplifying and solving logarithmic expressions.

Example: Simplify log₂8 + log₂4.

Solution: Using the product rule: log₂(8 * 4) = log₂32 = 5.

• Solving Logarithmic Equations: Many problems require solving for 'x' in logarithmic equations. This often involves using the definition of a logarithm or the properties of logarithms to simplify the equation.

Example: Solve for x in log₃(x + 2) = 2.

Solution: Rewrite in exponential form: 3² = x + 2. Solving for x gives x = 7.

• Change of Base Formula: The change of base formula allows you to convert a logarithm from one base to another, which is helpful when using calculators.

Section 3: Applications and Connections

Connecting Exponential and Logarithmic Functions:

Many problems require understanding the inverse relationship between exponential and logarithmic functions. Being able to switch between logarithmic and exponential forms is essential.

Real-World Applications:

This unit often concludes with applying these functions to model real-world scenarios such as compound interest calculations, radioactive decay estimations, and decibel measurements (sound intensity). These problems emphasize the practical uses of these functions.

Addressing Specific Questions

To get more specific answers, please provide the actual questions from your Unit 4 assignment. Include the problem statements and any relevant diagrams or data. I can then provide tailored solutions and explanations. Remember to always check your work against your textbook or class notes for the most accurate solutions.