Solving 2ln(4x) = 2ln(8) A Step-by-Step Guide

The question at hand is finding the true solution to the logarithmic equation 2ln(4x) = 2ln(8). Logarithmic equations can sometimes be tricky, so it's crucial to follow the correct steps to arrive at the accurate answer. In this comprehensive guide, we will meticulously walk through the solution process, ensuring each step is clearly explained and justified. We'll explore the properties of logarithms, demonstrate how to isolate the variable x, and verify the solution to guarantee its validity. By the end of this article, you'll not only have the answer but also a solid understanding of the methods used to solve such equations. The correct answer among the provided options (A. x = -4, B. x = -2, C. x = 2, D. x = 4) is essential, but the journey to the solution is equally important for grasping the underlying concepts.

Breaking Down the Equation: A Step-by-Step Solution

To effectively solve the logarithmic equation 2ln(4x) = 2ln(8), we need to employ several key properties of logarithms and algebraic manipulation techniques. Our objective is to isolate x and determine its value. The step-by-step solution is as follows:

Step 1: Divide Both Sides by 2

The initial equation is 2ln(4x) = 2ln(8). To simplify, we can divide both sides of the equation by 2. This operation maintains the equality and eliminates the coefficient of 2 from the logarithmic terms. Dividing both sides by 2 yields:

ln(4x) = ln(8)

This simplifies the equation, making it easier to work with. The next step involves utilizing the properties of logarithms to further isolate x.

Step 2: Exponentiate Both Sides Using the Base of the Natural Logarithm

The natural logarithm, denoted as ln, has a base of e (Euler's number, approximately 2.71828). To eliminate the natural logarithm, we can exponentiate both sides of the equation using e as the base. This is based on the property that e raised to the power of ln(a) is simply a. Thus, we have:

e^(ln(4x)) = e^(ln(8))

Using the property mentioned above, we simplify this to:

4x = 8

Now we have a much simpler algebraic equation to solve.

Step 3: Isolate x by Dividing Both Sides by 4

To isolate x, we divide both sides of the equation 4x = 8 by 4. This gives us:

x = 8 / 4

Simplifying the right side of the equation:

x = 2

Therefore, the solution to the equation 2ln(4x) = 2ln(8) is x = 2. However, we must always verify our solution to ensure it is valid, especially when dealing with logarithmic equations.

Verification of the Solution: Ensuring Validity

In the context of logarithmic equations, it is crucial to verify the solution because the domain of logarithmic functions is restricted to positive arguments. This means that the expression inside the logarithm must be greater than zero. To verify our solution x = 2, we substitute this value back into the original equation:

2ln(4x) = 2ln(8)

Substitute x = 2:

2ln(4 * 2) = 2ln(8)

Simplify:

2ln(8) = 2ln(8)

The equation holds true, confirming that x = 2 is indeed a valid solution. If substituting the value of x led to the logarithm of a non-positive number or an inequality, it would indicate that the solution is extraneous and not valid.

Common Pitfalls and How to Avoid Them

When solving logarithmic equations, several common pitfalls can lead to incorrect solutions. Understanding these pitfalls and how to avoid them is crucial for accurate problem-solving. Here are some common mistakes:

1. Ignoring the Domain of Logarithmic Functions: Logarithmic functions are only defined for positive arguments. Always check that the value of x you obtain does not result in taking the logarithm of a negative number or zero. If it does, the solution is extraneous.
2. Incorrectly Applying Logarithmic Properties: Logarithmic properties, such as the power rule, product rule, and quotient rule, must be applied correctly. A mistake in applying these rules can lead to an incorrect simplification and solution.
3. Forgetting to Verify Solutions: Even if you follow all the steps correctly, it is essential to verify your solution by substituting it back into the original equation. This ensures that the solution is valid and not extraneous.
4. Misunderstanding the Base of the Logarithm: The base of the logarithm is crucial. For example, ln refers to the natural logarithm with a base of e, while log without a specified base is often assumed to be base 10. Using the wrong base in calculations can lead to errors.
5. Incorrectly Exponentiating: When exponentiating both sides of an equation to eliminate a logarithm, ensure you use the correct base. For natural logarithms (ln), use e as the base. For common logarithms (base 10), use 10 as the base.

By being mindful of these common pitfalls and consistently verifying your solutions, you can improve your accuracy in solving logarithmic equations.

Alternative Methods for Solving Logarithmic Equations

While the step-by-step method outlined above is a standard approach for solving logarithmic equations, there are alternative methods that can be used depending on the specific equation. Understanding these different approaches can provide flexibility in problem-solving.

Method 1: Combining Logarithms

If the equation involves multiple logarithmic terms, one approach is to combine them into a single logarithm using logarithmic properties. For example, the product rule states that ln(a) + ln(b) = ln(ab), and the quotient rule states that ln(a) - ln(b) = ln(a/b). Applying these rules can simplify the equation before exponentiating.

Method 2: Converting to Exponential Form

Another method is to directly convert the logarithmic equation into its equivalent exponential form. For example, the equation ln(x) = y is equivalent to e^y = x. This can be particularly useful when the equation is in a simple logarithmic form.

Method 3: Graphical Solutions

In some cases, logarithmic equations can be solved graphically by plotting the functions on both sides of the equation and finding the points of intersection. This method is especially useful for equations that are difficult to solve algebraically.

Method 4: Numerical Methods

For complex logarithmic equations that cannot be solved analytically, numerical methods such as the Newton-Raphson method or iterative techniques can be used to approximate the solutions.

By familiarizing yourself with these alternative methods, you can choose the most efficient approach for solving different types of logarithmic equations.

Real-World Applications of Logarithmic Equations

Logarithmic equations are not just theoretical mathematical constructs; they have numerous real-world applications in various fields. Understanding these applications can provide a deeper appreciation for the importance of logarithmic functions and equations.

1. Chemistry

In chemistry, logarithmic scales are used to represent quantities that vary over a wide range. For example, the pH scale, which measures the acidity or alkalinity of a solution, is logarithmic. The pH is defined as the negative logarithm (base 10) of the hydrogen ion concentration: pH = -log[H+].

2. Physics

In physics, logarithmic scales are used to measure sound intensity (decibels) and earthquake magnitude (the Richter scale). The decibel scale is based on the logarithm of the ratio of sound intensities, while the Richter scale measures the logarithm of the amplitude of seismic waves.

3. Finance

In finance, logarithmic functions are used to model compound interest and the growth of investments. The logarithmic growth model is particularly useful for analyzing long-term financial trends.

4. Computer Science

In computer science, logarithms are used to analyze the efficiency of algorithms. The time complexity of many algorithms is expressed in logarithmic terms (e.g., O(log n)), which indicates that the algorithm's performance scales logarithmically with the size of the input.

5. Biology

In biology, logarithmic scales are used to model population growth and radioactive decay. The exponential growth or decay of a population or radioactive substance can be described using logarithmic equations.

6. Data Analysis

In data analysis, logarithmic transformations are often applied to data to stabilize variance and make it more suitable for statistical analysis. Logarithmic scaling can also reveal patterns that are not apparent in the original data.

The widespread use of logarithmic equations in these diverse fields underscores their fundamental importance in mathematical modeling and problem-solving.

Conclusion: Mastering Logarithmic Equations

In summary, the true solution to the equation 2ln(4x) = 2ln(8) is x = 2. We arrived at this solution by dividing both sides of the equation by 2, exponentiating using the base of the natural logarithm, and isolating x. Crucially, we verified the solution to ensure its validity within the domain of logarithmic functions. Solving logarithmic equations requires a solid understanding of logarithmic properties and careful application of algebraic techniques. Common pitfalls, such as ignoring the domain of logarithmic functions or misapplying logarithmic properties, can lead to incorrect answers. By avoiding these pitfalls and consistently verifying solutions, one can master logarithmic equations. Furthermore, understanding alternative methods for solving logarithmic equations and recognizing their real-world applications can deepen one's appreciation for their importance in various fields. Through practice and a clear understanding of the underlying principles, anyone can confidently tackle logarithmic equations and their applications.