Unit 7 Exponential And Logarithmic Functions Homework 4 Answers

Unit 7: Exponential and Logarithmic Functions - Homework 4 Answers & Comprehensive Guide

This comprehensive guide delves into the solutions and explanations for a typical Unit 7, Homework 4 assignment focusing on exponential and logarithmic functions. We'll cover a wide range of problems, providing detailed steps and insights to solidify your understanding of these crucial mathematical concepts. Remember, understanding the why behind the solution is just as important as getting the correct answer. This guide aims to achieve both.

Understanding Exponential Functions

Before diving into the homework problems, let's refresh our understanding of exponential functions. An exponential function is a function of the form:

f(x) = a^x

where 'a' is a positive constant called the base, and 'x' is the exponent. The key characteristic of exponential functions is their rapid growth or decay. If 'a' > 1, the function exhibits exponential growth; if 0 < 'a' < 1, it exhibits exponential decay.

Key Properties of Exponential Functions:

• Domain: All real numbers (-∞, ∞)
• Range: (0, ∞) (Always positive)
• x-intercept: None (unless a modification is applied, such as adding a constant)
• y-intercept: (0, 1) (If no constant is added)
• Asymptote: The x-axis (y = 0) acts as a horizontal asymptote.

Understanding Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are defined as:

y = log_a(x)

This equation is equivalent to:

a^y = x

where 'a' is the base (again, a positive constant not equal to 1), 'x' is the argument (must be positive), and 'y' is the logarithm. Logarithms answer the question: "To what power must I raise the base ('a') to get the argument ('x')?"

Key Properties of Logarithmic Functions:

• Domain: (0, ∞) (Argument must be positive)
• Range: All real numbers (-∞, ∞)
• x-intercept: (1, 0)
• y-intercept: None
• Asymptote: The y-axis (x = 0) acts as a vertical asymptote.

Homework 4 Problem Examples & Solutions

Let's tackle some common problems found in a typical Unit 7, Homework 4 assignment. We'll use specific examples to illustrate the concepts and problem-solving techniques. Remember to always show your work clearly; this aids both understanding and grading.

Problem 1: Evaluating Exponential Expressions

Question: Evaluate 2³, 10⁰, and (1/2)^-2.

Solution:

• 2³ = 2 * 2 * 2 = 8
• 10⁰ = 1 (Anything raised to the power of 0 is 1, except for 0⁰ which is undefined.)
• (1/2)^-2 = 2² = 4 (Recall that a^-n = 1/aⁿ)

Problem 2: Solving Exponential Equations

Question: Solve for x: 3^x = 81

Solution: We need to express 81 as a power of 3:

81 = 3⁴

Therefore, 3^x = 3⁴. Since the bases are equal, we can equate the exponents:

x = 4

Problem 3: Solving Logarithmic Equations

Question: Solve for x: log₂(x) = 3

Solution: By definition of a logarithm, this equation is equivalent to:

2³ = x

Therefore, x = 8

Problem 4: Graphing Exponential and Logarithmic Functions

Question: Sketch the graph of f(x) = 2^x and its inverse, g(x) = log₂(x).

Solution:

To graph f(x) = 2^x, you would plot points such as (0,1), (1,2), (2,4), (-1, 1/2), etc. The graph will show exponential growth, approaching the x-axis asymptotically.

The graph of g(x) = log₂(x) will be the reflection of f(x) = 2^x across the line y = x. It will show logarithmic growth, approaching the y-axis asymptotically. Key points include (1,0), (2,1), (4,2), (1/2, -1), etc. You should be comfortable using graphing calculators or software for accurate plotting and visualization.

Problem 5: Applications of Exponential and Logarithmic Functions

Question: A population of bacteria doubles every hour. If the initial population is 1000, what will the population be after 3 hours?

Solution: This is an example of exponential growth. The formula for exponential growth is:

P(t) = P₀ * 2^t

where P(t) is the population at time t, P₀ is the initial population, and t is the time in hours.

In this case, P₀ = 1000. After 3 hours, the population will be:

P(3) = 1000 * 2³ = 1000 * 8 = 8000

Problem 6: Using Properties of Logarithms

Question: Simplify log₁₀(100) + log₁₀(1000).

Solution: We use the property that log_a(xy) = log_a(x) + log_a(y):

log₁₀(100) + log₁₀(1000) = log₁₀(100 * 1000) = log₁₀(100000) = log₁₀(10⁵) = 5

Problem 7: Change of Base Formula

Question: Express log₂(5) in terms of base 10 logarithms.

Solution: Use the change of base formula:

log_a(x) = log_b(x) / log_b(a)

So, log₂(5) = log₁₀(5) / log₁₀(2)

Problem 8: Solving More Complex Exponential and Logarithmic Equations

Question: Solve for x: 2^2x + 2^x - 6 = 0

Solution: This equation can be solved by substitution. Let y = 2^x. Then the equation becomes:

y² + y - 6 = 0

This is a quadratic equation that can be factored:

(y + 3)(y - 2) = 0

This gives two possible solutions for y: y = -3 and y = 2. Since y = 2^x, y must be positive. Therefore, y = 2.

Substituting back: 2^x = 2, which means x = 1.

Problem 9: Word Problems Involving Exponential Decay

Question: The half-life of a radioactive substance is 5 years. If you start with 10 grams, how much will remain after 15 years?

Solution: The formula for exponential decay is:

A(t) = A₀ * (1/2)^t/h

where A(t) is the amount remaining after time t, A₀ is the initial amount, t is the time elapsed, and h is the half-life.

In this case, A₀ = 10 grams, h = 5 years, and t = 15 years. Therefore:

A(15) = 10 * (1/2)^15/5 = 10 * (1/2)³ = 10 * (1/8) = 1.25 grams

Beyond the Homework: Further Exploration

This guide provided solutions and explanations for a representative set of problems from a typical Unit 7, Homework 4 assignment on exponential and logarithmic functions. However, mastering these concepts requires more than just solving problems; it necessitates a deep understanding of their underlying principles and applications. Here are some avenues for further exploration:

• Practice more problems: Seek out additional exercises in your textbook or online resources.
• Explore real-world applications: Investigate how exponential and logarithmic functions are used in fields like finance (compound interest), biology (population growth and decay), and physics (radioactive decay).
• Use technology: Utilize graphing calculators or software to visualize functions and gain a better intuitive understanding of their behavior.
• Work with different bases: Practice problems involving logarithms with bases other than 10 and e (the natural logarithm base).
• Understand the relationship between exponential and logarithmic functions: Remember that they are inverses of each other, a key concept for solving many problems.

By diligently working through problems, exploring applications, and utilizing available resources, you can solidify your understanding of exponential and logarithmic functions and excel in your studies. Remember that consistent effort and a focus on understanding the underlying principles are essential for true mastery.