Given Log₂3 = X For A Real Number X, Where 1 < X < 2, Determine The Correct Statement And Its Justification.

In the realm of mathematics, logarithms serve as a powerful tool for unraveling exponential relationships. They provide a way to express exponents in a more manageable form, enabling us to solve complex equations and gain deeper insights into mathematical structures. This article delves into the intricacies of logarithms, focusing on the specific expression log₂3 = x, where x is a real number between 1 and 2. We will explore the fundamental concepts of logarithms, examine the properties that govern their behavior, and ultimately justify the assertion that 1 < x < 2.

What are Logarithms?

At its core, a logarithm answers the question: "To what power must we raise a base to obtain a specific number?" In the expression logₐb = c, 'a' represents the base, 'b' is the number we want to obtain, and 'c' is the exponent to which we must raise 'a'. In simpler terms, logₐb = c is equivalent to aᶜ = b. For example, log₂8 = 3 because 2³ = 8. Here, the base is 2, the number we want to obtain is 8, and the exponent is 3.

Logarithms are intimately connected to exponential functions. They are essentially the inverse operations of exponentiation. Just as subtraction undoes addition and division undoes multiplication, logarithms undo exponentiation. This inverse relationship is crucial for solving exponential equations and understanding the behavior of logarithmic functions.

Exploring log₂3 = x

Now, let's focus on the expression log₂3 = x. This expression asks the question: "To what power must we raise 2 to obtain 3?" The value of x represents this power. To understand the nature of x, we can rewrite the logarithmic equation in its exponential form: 2ˣ = 3. This equation states that 2 raised to the power of x equals 3. Our goal is to determine the value of x and demonstrate that it lies between 1 and 2.

Why 1 < x < 2?

To establish that 1 < x < 2, we can use the properties of exponents and inequalities. We know that 2¹ = 2 and 2² = 4. Since 3 lies between 2 and 4, it follows that the exponent x, which satisfies 2ˣ = 3, must lie between 1 and 2. This is because the exponential function 2ˣ is an increasing function. As x increases, 2ˣ also increases. Therefore, if 2 < 3 < 4, then the corresponding exponents must satisfy 1 < x < 2.

A Deeper Dive into the Reasoning

Let's break down the reasoning step by step:

1. Establish the bounds: We start by identifying two powers of 2 that bound 3. We know that 2¹ = 2 and 2² = 4. Since 2 < 3 < 4, we have established the lower and upper bounds for 3 in terms of powers of 2.
2. Apply the increasing nature of exponential functions: The function f(x) = 2ˣ is an increasing function. This means that as x increases, f(x) also increases. In other words, if a < b, then 2ᵃ < 2ᵇ. This property is crucial for our argument.
3. Connect the inequalities: Since 2 < 3 < 4, we can write this as 2¹ < 3 < 2². Now, we introduce the exponent x, which satisfies 2ˣ = 3. Substituting this into the inequality, we get 2¹ < 2ˣ < 2².
4. Deduce the bounds for x: Because the exponential function 2ˣ is increasing, we can deduce that the exponents must also satisfy the same inequality. Therefore, 1 < x < 2. This confirms that the value of x, which is log₂3, lies between 1 and 2.

Properties of Logarithms

Understanding the properties of logarithms is essential for manipulating logarithmic expressions and solving logarithmic equations. Here are some key properties:

• Product Rule: logₐ(mn) = logₐm + logₐn
• Quotient Rule: logₐ(m/n) = logₐm - logₐn
• Power Rule: logₐ(mᵖ) = p * logₐm
• Change of Base Formula: logₐb = logₓb / logₓa (where x is any valid base)

These properties allow us to simplify complex logarithmic expressions, combine or separate logarithmic terms, and change the base of a logarithm. They are fundamental tools for working with logarithms in various mathematical contexts.

Applying Logarithmic Properties to log₂3

While the properties of logarithms are not directly used to prove that 1 < log₂3 < 2, they can be used to approximate the value of log₂3 or to relate it to other logarithmic expressions. For instance, we can use the change of base formula to express log₂3 in terms of the natural logarithm (base e) or the common logarithm (base 10):

log₂3 = ln(3) / ln(2) ≈ 1.585

log₂3 = log₁₀(3) / log₁₀(2) ≈ 1.585

These approximations confirm that log₂3 is indeed between 1 and 2.

Conclusion

In conclusion, the expression log₂3 = x, where x is a real number, implies that 2ˣ = 3. By understanding the increasing nature of exponential functions and bounding 3 between powers of 2, we can confidently assert that 1 < x < 2. This exploration highlights the fundamental concepts of logarithms and their relationship to exponential functions. The properties of logarithms, while not directly used in this proof, provide valuable tools for manipulating and approximating logarithmic expressions. The ability to work with logarithms is essential for various mathematical applications, including solving equations, modeling growth and decay, and analyzing data.

Given the equation log₂3 = x, where x is a real number such that 1 < x < 2, we aim to identify the correct statement regarding the value of x. This involves understanding the properties of logarithms and their relationship with exponential functions. As established previously, the equation log₂3 = x is equivalent to 2ˣ = 3. The condition 1 < x < 2 arises from the fact that 2¹ = 2 and 2² = 4, and since 3 lies between 2 and 4, x must lie between 1 and 2. We will analyze various possible statements and justify why a specific statement is true while others are false.

Analyzing Potential Statements

To determine the correct statement, we need to consider different ways to express the relationship between log₂3 and x. This might involve comparing x to specific values, establishing inequalities, or relating x to other logarithmic expressions. Let's consider some potential statements:

• Statement A: x is closer to 1 than to 2.
• Statement B: x is exactly 1.5.
• Statement C: x is closer to 2 than to 1.

To evaluate these statements, we need to consider the value of log₂3 more precisely. We already know that 1 < x < 2, but we need to refine this estimate to determine which of the statements is most accurate.

Evaluating Statement A: x is closer to 1 than to 2.

This statement suggests that log₂3 is closer to 1 than to 2. This would imply that 2ˣ = 3 is closer to 2¹ = 2 than to 2² = 4. To evaluate this, we can consider the midpoint between 2 and 4, which is 3. If 2ˣ = 3 is closer to 2 than to 4, then x should be closer to 1 than to 2. However, without a precise value for x, it's difficult to definitively say whether this statement is true or false.

Evaluating Statement B: x is exactly 1.5.

This statement claims that log₂3 = 1.5. If this were true, then 2¹⁵ should equal 3. Let's check this: 2¹⁵ = 2^(3/2) = √(2³) = √8 ≈ 2.828. Since 2.828 is not equal to 3, this statement is false. Therefore, x is not exactly 1.5.

Evaluating Statement C: x is closer to 2 than to 1.

This statement suggests that log₂3 is closer to 2 than to 1. This would imply that 2ˣ = 3 is closer to 2² = 4 than to 2¹ = 2. This seems more likely than Statement A, given that 3 is closer to 4 than it is to 2. To confirm this, we need a more accurate estimate of log₂3.

Obtaining a More Accurate Estimate

We can use a calculator or a numerical method to approximate the value of log₂3. Using a calculator, we find that log₂3 ≈ 1.585. This value is indeed closer to 1.5 than to 1 or 2. However, to definitively say whether it's closer to 1 or 2, we can compare the distances:

• Distance from x to 1: |1.585 - 1| = 0.585
• Distance from x to 2: |1.585 - 2| = 0.415

Since 0.415 < 0.585, log₂3 is closer to 2 than to 1. This confirms that Statement C is the correct statement.

Justifying the Correct Statement

The justification for Statement C (x is closer to 2 than to 1) lies in the fact that the exponential function 2ˣ increases more rapidly as x increases. Since 3 is closer to 4 (2²) than it is to 2 (2¹), the corresponding value of x (log₂3) must be closer to 2 than to 1. We have also verified this by approximating the value of log₂3 and comparing its distances from 1 and 2.

Analyzing Potential Justifications

Now, let's consider potential justifications for Statement C. These justifications should provide a logical explanation for why log₂3 is closer to 2 than to 1. Here are some possible justifications:

1. 3 is closer to 4 than to 2.
2. The function 2ˣ increases at a decreasing rate.
3. 1.5 is the average of 1 and 2.

Let's analyze each justification:

1. 3 is closer to 4 than to 2: This is the most direct and relevant justification. Since 2¹ = 2 and 2² = 4, and 3 is closer to 4 than to 2, it logically follows that log₂3 is closer to 2 than to 1. This justification aligns with the increasing nature of the exponential function.
2. The function 2ˣ increases at a decreasing rate: This statement is incorrect. The exponential function 2ˣ increases at an increasing rate. As x increases, the rate of change of 2ˣ also increases. This is a fundamental property of exponential functions.
3. 1.5 is the average of 1 and 2: This statement is true, but it doesn't provide a valid justification for why log₂3 is closer to 2 than to 1. The average of 1 and 2 is indeed 1.5, but this doesn't imply that log₂3 should be closer to 2. It only suggests that 1.5 is the midpoint between 1 and 2.

Conclusion: The Correct Statement and Justification

Therefore, the correct statement is C: x is closer to 2 than to 1, and the correct justification is 1: 3 is closer to 4 than to 2. This combination provides a clear and logical explanation for why log₂3 lies closer to 2 than to 1. The increasing nature of the exponential function 2ˣ and the proximity of 3 to 4 are the key factors in this determination.

This comprehensive analysis demonstrates the importance of understanding the properties of logarithms and exponential functions when evaluating mathematical statements and justifications. By carefully considering the relationships between these concepts, we can arrive at accurate conclusions and provide sound reasoning for our assertions.

Therefore, the final answer is Statement C with Justification 1.