Complete The Table For Logarithmic Function Y = Log₃(x - 2)

Introduction to Logarithmic Functions

In mathematics, logarithmic functions are the inverses of exponential functions. Understanding logarithmic functions is crucial in various fields, including mathematics, physics, engineering, and computer science. They help us solve equations where the unknown is an exponent and model phenomena that exhibit exponential growth or decay. In this comprehensive guide, we will delve into the logarithmic function y = log₃(x - 2), exploring its properties, evaluating its values at specific points, and completing a table to understand its behavior.

At its core, a logarithmic function answers the question: "To what power must we raise the base to obtain a certain number?" The general form of a logarithmic function is y = logₐ(x), where a is the base, x is the argument, and y is the exponent. The base a must be a positive number not equal to 1. The argument x must be a positive number. The value of y represents the power to which the base a must be raised to equal x. The logarithmic function y = log₃(x - 2) is a specific example where the base is 3 and the argument is (x - 2). This means we are looking for the power to which we must raise 3 to obtain (x - 2). The function is defined only for values of x greater than 2, because the argument (x - 2) must be positive. This restriction is crucial to understanding the domain and behavior of the function.

Understanding logarithmic functions requires a solid grasp of exponential functions and their inverse relationship. The exponential function y = aˣ represents the value obtained by raising the base a to the power of x. The logarithmic function y = logₐ(x), on the other hand, asks for the value of x given the base a and the result y. These two functions are intimately linked, and one can be used to solve for the other. The function y = log₃(x - 2) is a transformation of the basic logarithmic function y = log₃(x). The horizontal shift by 2 units to the right is caused by the subtraction of 2 from the argument x. This shift affects the domain and the vertical asymptote of the function. The vertical asymptote of the function y = log₃(x) is the y-axis (x = 0), while the vertical asymptote of the function y = log₃(x - 2) is the line x = 2. This means that the function y = log₃(x - 2) is not defined for x ≤ 2. The domain of the function is (2, ∞), and its range is (-∞, ∞). The function increases monotonically, meaning that as x increases, y also increases. The rate of increase, however, slows down as x becomes larger. This behavior is characteristic of logarithmic functions and is important in modeling phenomena such as the decay of radioactive substances and the growth of populations under limited resources.

Evaluating the Logarithmic Function

To complete the table for the function y = log₃(x - 2), we need to evaluate the function for the given values of x. This involves substituting each x-value into the function and calculating the corresponding y-value. The process requires understanding the definition of logarithms and the properties of exponents. It also involves careful arithmetic and algebraic manipulation. The goal is to find the value of y that satisfies the equation 3ʸ = x - 2. This can be done by expressing (x - 2) as a power of 3 and then equating the exponents. Alternatively, one can use the change of base formula to convert the logarithm to a more convenient base, such as the natural logarithm or the common logarithm, and then use a calculator to approximate the value. The evaluation process also helps to reinforce the concept of logarithms and their relationship to exponents. It allows us to see how the function behaves for different values of x and how the y-value changes as x varies. This understanding is essential for graphing the function, analyzing its properties, and applying it to real-world problems. Each evaluation provides a data point that can be used to plot the graph of the function and to make predictions about its behavior.

Let's start by evaluating the function for x = 19/9. Substituting this value into the function, we get:

y = log₃(19/9 - 2)

First, simplify the expression inside the logarithm:

19/9 - 2 = 19/9 - 18/9 = 1/9

Now, the equation becomes:

y = log₃(1/9)

We need to find the power to which we must raise 3 to obtain 1/9. Since 1/9 = 3⁻², we have:

y = log₃(3⁻²)

Therefore:

y = -2

Next, let's evaluate the function for x = 7:

y = log₃(7 - 2)

Simplify the expression inside the logarithm:

y = log₃(5)

This value is not an integer, so we cannot simplify it further without using a calculator. We can express it as log base 3 of 5, which is approximately 1.465. Therefore:

y ≈ 1.465

Now, let's evaluate the function for x = 3:

y = log₃(3 - 2)

Simplify the expression inside the logarithm:

y = log₃(1)

The logarithm of 1 to any base is always 0, so:

y = 0

Finally, let's evaluate the function for x = 5:

y = log₃(5 - 2)

Simplify the expression inside the logarithm:

y = log₃(3)

The logarithm of a number to the same base is always 1, so:

y = 1

Completing the Table

Now that we have evaluated the function for all given values of x, we can complete the table:

This table provides a clear picture of how the function y = log₃(x - 2) behaves for different values of x. It shows that the function increases as x increases, but the rate of increase slows down as x becomes larger. The table also highlights the importance of the domain of the function, which is (2, ∞). The function is not defined for values of x less than or equal to 2, and this is reflected in the behavior of the function as x approaches 2. The table can be used to plot the graph of the function and to make predictions about its behavior for other values of x. It also serves as a valuable tool for understanding the properties of logarithmic functions and their applications in various fields.

Properties of Logarithmic Functions

Logarithmic functions possess several important properties that make them powerful tools in mathematics and other disciplines. These properties govern how logarithms behave under different operations and provide a means for simplifying complex expressions. Understanding these properties is crucial for solving logarithmic equations, graphing logarithmic functions, and applying them to real-world problems. The key properties include the product rule, the quotient rule, the power rule, and the change of base formula. Each of these properties has a unique role in manipulating logarithmic expressions and can be used to transform them into more manageable forms. The properties are derived from the corresponding properties of exponents, reflecting the inverse relationship between logarithms and exponential functions. By mastering these properties, one can gain a deeper understanding of logarithmic functions and their applications.

The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is expressed as logₐ(mn) = logₐ(m) + logₐ(n). This property is particularly useful for simplifying expressions where the argument of the logarithm is a product. For example, log₂(8 * 4) can be simplified to log₂(8) + log₂(4), which is 3 + 2 = 5. This property allows us to break down complex logarithms into simpler components, making them easier to evaluate. The product rule is a direct consequence of the property of exponents that states aˣ * aʸ = a⁽ˣ⁺ʸ⁾. The logarithm essentially reverses this operation, transforming multiplication into addition.

The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Mathematically, this is expressed as logₐ(m/n) = logₐ(m) - logₐ(n). This property is useful for simplifying expressions where the argument of the logarithm is a fraction. For example, log₂(16/2) can be simplified to log₂(16) - log₂(2), which is 4 - 1 = 3. Similar to the product rule, the quotient rule allows us to simplify complex logarithms into simpler components. The quotient rule is derived from the property of exponents that states aˣ / aʸ = a⁽ˣ⁻ʸ⁾. The logarithm again reverses this operation, transforming division into subtraction.

The power rule states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. Mathematically, this is expressed as logₐ(mⁿ) = n * logₐ(m). This property is particularly useful for simplifying expressions where the argument of the logarithm is raised to a power. For example, log₂(4³) can be simplified to 3 * log₂(4), which is 3 * 2 = 6. The power rule is a powerful tool for manipulating logarithms and is often used in solving logarithmic equations. The power rule is a consequence of the property of exponents that states (aˣ)ⁿ = a⁽ˣⁿ⁾. The logarithm again reverses this operation, transforming exponentiation into multiplication.

The change of base formula allows us to convert a logarithm from one base to another. This is particularly useful when we need to evaluate a logarithm with a base that is not readily available on a calculator. The formula is expressed as logₐ(b) = logₓ(b) / logₓ(a), where x is the new base. The new base x can be any positive number not equal to 1, but the most common choices are 10 (common logarithm) and e (natural logarithm). For example, to evaluate log₃(5), we can use the change of base formula to convert it to a common logarithm: log₃(5) = log₁₀(5) / log₁₀(3). Using a calculator, we can find that log₁₀(5) ≈ 0.699 and log₁₀(3) ≈ 0.477, so log₃(5) ≈ 0.699 / 0.477 ≈ 1.465. The change of base formula is essential for working with logarithms in different bases and for evaluating logarithms using calculators or computer software.

Conclusion

In this comprehensive exploration, we have successfully evaluated the logarithmic function y = log₃(x - 2) for various values of x and completed the table. This exercise has provided valuable insights into the behavior of logarithmic functions and their properties. Understanding logarithmic functions is essential for solving complex mathematical problems and for modeling real-world phenomena. The properties of logarithms, such as the product rule, quotient rule, power rule, and change of base formula, are powerful tools for simplifying expressions and solving equations. By mastering these concepts, one can gain a deeper appreciation for the role of logarithmic functions in mathematics and its applications. The logarithmic function y = log₃(x - 2) is a fundamental example that illustrates the key characteristics of logarithms, including their domain, range, and relationship to exponential functions. The ability to evaluate and manipulate logarithmic functions is a valuable skill for students and professionals in various fields.