Solving $4 \ln (x-5)=8$ For X A Step By Step Guide

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Understanding the Problem

In this mathematical problem, our primary goal is to determine the value of x that satisfies the given equation: 4ln(x-5) = 8. This equation involves a natural logarithm, which is a logarithm to the base e, where e is an irrational number approximately equal to 2.71828. Solving for x requires us to isolate x on one side of the equation by systematically unwinding the mathematical operations applied to it. This involves using the properties of logarithms and exponential functions to reverse the logarithmic operation and eventually solve for x. The solution must be precise, avoiding any intermediate rounding, and the final answer should be rounded to the nearest tenth. This ensures both accuracy and practicality in the solution.

The process of solving this equation will involve several key steps. First, we will divide both sides of the equation by 4 to isolate the natural logarithm term. Next, we will use the property that the exponential function is the inverse of the natural logarithm, applying the exponential function to both sides of the equation to eliminate the logarithm. This will give us an equation in terms of x-5. Finally, we will add 5 to both sides of the equation to solve for x. It is crucial to verify that the solution we obtain is valid by ensuring that the argument of the logarithm (x-5) is positive, as the logarithm of a non-positive number is undefined. This verification step is essential to ensure the mathematical integrity of our solution. Throughout this process, we aim to maintain precision and only round at the final step to minimize any potential errors.

Solving logarithmic equations like this requires a solid understanding of the properties of logarithms and exponential functions. Logarithms are essentially the inverse of exponential operations, and they help us solve equations where the variable is an exponent or within a logarithmic function. The natural logarithm, denoted as ln(x), is particularly important in calculus and many areas of science and engineering. Its inverse function, the exponential function e^x, allows us to move between logarithmic and exponential forms of equations. In this specific problem, we leverage this inverse relationship to eliminate the natural logarithm and isolate x. By carefully applying these principles and performing algebraic manipulations, we can find the value of x that satisfies the given equation, adhering to the specified rounding requirements and ensuring the validity of our solution.

Step-by-Step Solution

To solve the equation 4ln(x-5) = 8, we will proceed step-by-step, ensuring each operation is mathematically sound and follows the properties of logarithms and exponential functions.

  1. Isolate the Natural Logarithm: The first step is to isolate the natural logarithm term. We can do this by dividing both sides of the equation by 4:

    4ln(x-5) / 4 = 8 / 4

    This simplifies to:

    ln(x-5) = 2

  2. Eliminate the Natural Logarithm: To eliminate the natural logarithm, we use the property that the exponential function (base e) is the inverse of the natural logarithm. We apply the exponential function to both sides of the equation:

    e^(ln(x-5)) = e^2

    Since e raised to the power of the natural logarithm of a quantity is the quantity itself, we have:

    x - 5 = e^2

  3. Solve for x: Now, to solve for x, we add 5 to both sides of the equation:

    x = e^2 + 5

  4. Approximate the Value of x: We need to approximate the value of e^2 and then add 5. The value of e is approximately 2.71828, so e^2 is approximately:

    e^2 ≈ (2.71828)^2 ≈ 7.38906

    Adding 5 to this, we get:

    x ≈ 7.38906 + 5 ≈ 12.38906

  5. Round to the Nearest Tenth: Finally, we round the value of x to the nearest tenth:

    x ≈ 12.4

Therefore, the solution to the equation 4ln(x-5) = 8, rounded to the nearest tenth, is approximately 12.4. It is crucial to verify this solution by plugging it back into the original equation to ensure that it is valid. This step-by-step approach ensures accuracy and clarity in the solution process, making it easier to understand and replicate.

Verification of the Solution

To ensure the accuracy and validity of our solution, it's essential to verify the result x ≈ 12.4 by substituting it back into the original equation: 4ln(x-5) = 8. This verification process helps to catch any potential errors made during the solving process and confirms that our solution satisfies the given equation.

  1. Substitute x ≈ 12.4 into the Equation: We replace x with 12.4 in the original equation:

    4ln(12.4 - 5) = 8

  2. Simplify the Expression Inside the Logarithm: First, we simplify the expression inside the natural logarithm:

    12.4 - 5 = 7.4

    So the equation becomes:

    4ln(7.4) = 8

  3. Calculate the Natural Logarithm: Next, we find the natural logarithm of 7.4. Using a calculator, we find:

    ln(7.4) ≈ 2.00148

    Now the equation is:

    4 * 2.00148 ≈ 8

  4. Multiply and Compare: Multiplying 4 by 2.00148, we get:

    4 * 2.00148 ≈ 8.00592

    Comparing this result to the right-hand side of the original equation, we see:

    8.00592 ≈ 8

  5. Assess the Result: The calculated value, 8.00592, is very close to 8. This slight difference is due to rounding x to the nearest tenth. If we had used a more precise value for x before rounding, the result would have been even closer to 8. However, for practical purposes and given the initial rounding instruction, our solution is verified to be accurate.

Additionally, we must ensure that the argument of the logarithm (x-5) is positive. Since 12.4 - 5 = 7.4, which is positive, the logarithm is defined for our solution. This check is crucial because the logarithm of a non-positive number is undefined, and such a result would invalidate our solution.

In conclusion, by substituting x ≈ 12.4 back into the original equation and confirming that the result is approximately equal to 8, and by verifying that the argument of the logarithm is positive, we have successfully verified our solution. This rigorous verification process provides confidence in the accuracy and validity of our answer.

Common Mistakes to Avoid

When solving equations involving logarithms, it's crucial to be meticulous and avoid common pitfalls that can lead to incorrect solutions. Here are some frequent mistakes to watch out for:

  1. Incorrectly Applying the Order of Operations: One common mistake is not following the correct order of operations (PEMDAS/BODMAS). In our problem, 4ln(x-5) = 8, it's essential to first isolate the logarithm term before attempting to eliminate it. This means dividing both sides by 4 before dealing with the natural logarithm. Failing to do so can lead to incorrect algebraic manipulations and a wrong answer.

  2. Misunderstanding Logarithmic Properties: Logarithms have specific properties that must be correctly applied. For instance, the inverse relationship between the natural logarithm and the exponential function is crucial for solving this equation. The property e^(ln(x)) = x allows us to eliminate the natural logarithm. Misapplying this or other logarithmic properties can result in an incorrect solution. For example, incorrectly distributing the logarithm or exponentiating only part of an equation are common errors.

  3. Forgetting to Check the Domain of the Logarithm: Logarithmic functions are only defined for positive arguments. This means that the expression inside the logarithm must be greater than zero. In our case, x - 5 must be greater than 0. After solving for x, it is vital to check that the solution satisfies this condition. Forgetting this step can lead to extraneous solutions that do not actually satisfy the original equation. For example, if we had obtained a solution less than or equal to 5, it would be invalid.

  4. Rounding Intermediate Values: Rounding intermediate values during the solution process can introduce errors that compound and lead to an inaccurate final answer. To maintain precision, it's best to avoid rounding until the very last step. In our problem, approximating e^2 prematurely can lead to a slightly different final result. Keeping the exact value until the end and only rounding the final answer to the specified decimal place ensures greater accuracy.

  5. Algebraic Errors: Simple algebraic errors, such as incorrect addition, subtraction, multiplication, or division, can derail the solution process. It's important to double-check each step to ensure accuracy. For example, when isolating x, adding 5 to both sides of the equation must be done correctly to avoid mistakes.

By being aware of these common mistakes and diligently checking each step, you can increase your chances of solving logarithmic equations accurately and efficiently. Attention to detail and a solid understanding of logarithmic properties are key to success in these types of problems.

Conclusion

In summary, solving the equation 4ln(x-5) = 8 involves a series of steps that require a strong understanding of logarithmic properties and careful algebraic manipulation. The process begins by isolating the natural logarithm term, followed by using the exponential function to eliminate the logarithm, and finally, solving for x. Throughout this process, it's crucial to avoid common mistakes such as incorrectly applying the order of operations, misunderstanding logarithmic properties, neglecting the domain of the logarithm, rounding intermediate values, and making algebraic errors.

To recap the solution, we first divided both sides of the equation by 4 to isolate the natural logarithm, resulting in ln(x-5) = 2. Next, we applied the exponential function to both sides to eliminate the natural logarithm, giving us x - 5 = e^2. We then solved for x by adding 5 to both sides, yielding x = e^2 + 5. Approximating e^2 as 7.38906 and adding 5, we found x to be approximately 12.38906. Finally, we rounded this value to the nearest tenth, obtaining the solution x ≈ 12.4.

Verification of the solution is a critical step to ensure accuracy. By substituting x ≈ 12.4 back into the original equation, we confirmed that the result is approximately equal to 8, validating our solution. We also checked that the argument of the logarithm, x - 5, is positive, ensuring that our solution is within the domain of the logarithmic function.

Solving logarithmic equations like this is a fundamental skill in mathematics, with applications in various fields such as physics, engineering, and finance. The ability to accurately manipulate and solve these equations is essential for problem-solving and analytical thinking. By following a systematic approach, being mindful of common errors, and verifying the solution, one can confidently tackle logarithmic equations and achieve accurate results. The key to success lies in understanding the properties of logarithms, applying them correctly, and maintaining precision throughout the solution process.