Identifying The Equation With X=4 As The Solution

In the realm of mathematics, equations serve as powerful tools to model and solve a wide array of problems. Logarithmic equations, in particular, play a crucial role in various scientific and engineering applications. Solving these equations often involves identifying the value(s) of the variable that satisfy the given equation. In this article, we embark on a comprehensive exploration to determine which equation, among the provided options, has x = 4 as its solution. We will delve into the fundamental concepts of logarithms, their properties, and the techniques used to solve logarithmic equations. This journey will not only equip you with the knowledge to solve similar problems but also enhance your understanding of the underlying mathematical principles. Our primary focus will be on the following equations:

1. log₄(3x + 4) = 2
2. log₃(2x - 5) = 2
3. logₓ 64 = 4
4. logₓ 16 = 4

Each of these equations presents a unique challenge and requires a careful application of logarithmic properties to arrive at the solution. We will systematically analyze each equation, substituting x = 4 to verify if it satisfies the equation. Through this process, we will identify the equation that correctly yields x = 4 as its solution. Let's begin our exploration by revisiting the fundamental concepts of logarithms.

Before we dive into solving the equations, it is crucial to establish a firm understanding of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Mathematically, this relationship can be expressed as:

logₐ b = c if and only if aᶜ = b

Where:

• a is the base of the logarithm.
• b is the argument of the logarithm (the number whose logarithm is being taken).
• c is the logarithm itself (the exponent).

Understanding this fundamental relationship is key to solving logarithmic equations. It allows us to convert logarithmic expressions into their equivalent exponential forms, making them easier to manipulate and solve. For instance, the equation log₂ 8 = 3 can be rewritten in exponential form as 2³ = 8, which is a clear and straightforward statement. Now, let's explore some key properties of logarithms that will be instrumental in our quest to find the equation with x = 4 as the solution.

Logarithms possess several useful properties that simplify their manipulation and are crucial for solving logarithmic equations. Here are some of the most relevant properties for our task:

1. Product Rule: logₐ (mn) = logₐ m + logₐ n
   • This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. While not directly applicable in the given equations, it is a fundamental property worth noting.
2. Quotient Rule: logₐ (m/n) = logₐ m - logₐ n
   • Similar to the product rule, this rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.
3. Power Rule: logₐ (mᵖ) = p logₐ m
   • The power rule is particularly useful when dealing with exponents within logarithms. It states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number.
4. Change of Base Rule: logₓ b = logₐ b / logₐ x
   • The change of base rule allows us to convert logarithms from one base to another. This is especially helpful when dealing with logarithms with different bases.
5. logₐ a = 1
   • This property states that the logarithm of a number to the same base is always equal to 1. For example, log₂ 2 = 1, log₁₀ 10 = 1, and so on.
6. logₐ 1 = 0
   • The logarithm of 1 to any base is always equal to 0. This is because any number raised to the power of 0 is equal to 1.

These properties form the foundation for manipulating and solving logarithmic equations. As we proceed to analyze each equation, we will utilize these properties to simplify the expressions and determine whether x = 4 is a valid solution.

Now, let's delve into the first equation: log₄(3x + 4) = 2. To determine if x = 4 is a solution, we will substitute this value into the equation and verify if the equation holds true.

Substituting x = 4 into the equation, we get:

log₄(3(4) + 4) = 2

Simplifying the expression inside the logarithm:

log₄(12 + 4) = 2

log₄(16) = 2

Now, we need to check if this statement is true. Recall the fundamental relationship between logarithms and exponents: logₐ b = c if and only if aᶜ = b. Applying this to our equation, we can rewrite log₄(16) = 2 in exponential form as:

4² = 16

This statement is indeed true, as 4 squared is equal to 16. Therefore, x = 4 is a solution to the equation log₄(3x + 4) = 2.

Moving on to the second equation, log₃(2x - 5) = 2, we will again substitute x = 4 and check for validity.

Substituting x = 4 into the equation, we get:

log₃(2(4) - 5) = 2

Simplifying the expression inside the logarithm:

log₃(8 - 5) = 2

log₃(3) = 2

To verify this, we rewrite the equation in exponential form:

3² = 3

This statement is false, as 3 squared is equal to 9, not 3. Therefore, x = 4 is not a solution to the equation log₃(2x - 5) = 2.

Now, let's analyze the third equation: logₓ 64 = 4. Substituting x = 4, we get:

log₄ 64 = 4

Converting this to exponential form:

4⁴ = 64

Calculating 4 raised to the power of 4, we get:

4⁴ = 4 * 4 * 4 * 4 = 256

Since 256 is not equal to 64, the statement is false. Therefore, x = 4 is not a solution to the equation logₓ 64 = 4.

Finally, let's examine the fourth equation: logₓ 16 = 4. Substituting x = 4, we have:

log₄ 16 = 4

Converting this to exponential form:

4⁴ = 16

As we calculated earlier, 4⁴ = 256, which is not equal to 16. Therefore, the statement is false, and x = 4 is not a solution to the equation logₓ 16 = 4. However, there seems to be a mistake here. Let's correct the exponential form. The correct conversion to exponential form should be:

4⁴ = 16 is incorrect.

The correct exponential form should be:

x⁴ = 16

Substituting x = 4:

4⁴ = 16

This is still incorrect. The correct way to think about this is: What number raised to the power of 4 equals 16?

Let's rewrite the exponential equation:

x⁴ = 16

Take the fourth root of both sides:

x = ⁴√16

x = 2

Since x=4 is the question, let's see if we can manipulate:

logₓ 16 = 4

If x=4:

log₄ 16 = 4

Convert to exponential form:

4⁴ = 16

256=16 which is false.

Through our systematic analysis, we have examined each equation and determined whether x = 4 is a solution. Our findings are as follows:

• Equation 1: log₄(3x + 4) = 2 - x = 4 is a solution.
• Equation 2: log₃(2x - 5) = 2 - x = 4 is not a solution.
• Equation 3: logₓ 64 = 4 - x = 4 is not a solution.
• Equation 4: logₓ 16 = 4 - x = 4 is not a solution.

Therefore, the equation that has x = 4 as the solution is log₄(3x + 4) = 2. This exploration highlights the importance of understanding logarithmic properties and the relationship between logarithms and exponents in solving logarithmic equations. By carefully substituting and verifying, we can accurately identify the solutions to these equations.