Solve For X Step-by-Step Guide To Solving 4 + Ln(x-1) - 2 = 0
Introduction
In the realm of mathematics, solving equations is a fundamental skill. This article provides a step-by-step guide on how to solve the equation 4 + ln(x-1) - 2 = 0 for x. We will explore the properties of logarithms and algebraic manipulations required to isolate x and find its value. We will focus on providing clear explanations and avoiding rounding intermediate computations to ensure accuracy. The final answer will be rounded to the nearest hundredth.
Understanding the Equation
The equation we aim to solve is 4 + ln(x-1) - 2 = 0. It involves a natural logarithm, denoted as ln, which is the logarithm to the base e, where e is approximately 2.71828. The expression ln(x-1) indicates the natural logarithm of the quantity (x-1). To solve this equation, we need to isolate the logarithmic term and then use the properties of logarithms to solve for x. Before diving into the steps, it’s crucial to understand the domain of the logarithmic function. The natural logarithm ln(u) is defined only for u > 0. Therefore, in our case, (x-1) must be greater than 0, meaning x > 1. This condition is crucial because it determines the range of acceptable solutions.
Step-by-Step Solution
1. Simplify the Equation
First, we simplify the given equation by combining the constants: 4 + ln(x-1) - 2 = 0. By combining the constants 4 and -2, we get: ln(x-1) + 2 = 0. This simplified form makes the equation easier to work with. Our next goal is to isolate the logarithmic term on one side of the equation. This involves subtracting 2 from both sides of the equation. This algebraic manipulation is a standard technique used to isolate variables or functions in an equation.
2. Isolate the Logarithmic Term
To isolate the logarithmic term, we subtract 2 from both sides of the equation: ln(x-1) + 2 - 2 = 0 - 2. This simplifies to: ln(x-1) = -2. Now, the natural logarithm term is isolated, which sets the stage for using the properties of logarithms to solve for x. The equation ln(x-1) = -2 means that the natural logarithm of (x-1) is equal to -2. To solve for x, we need to undo the logarithm, which involves exponentiating both sides of the equation.
3. Exponentiate Both Sides
To remove the natural logarithm, we exponentiate both sides of the equation using the base e. This is because the exponential function with base e is the inverse of the natural logarithm. The equation ln(x-1) = -2 becomes: e^(ln(x-1)) = e^(-2). Using the property that e^(ln(u)) = u, we simplify the left side of the equation: x-1 = e^(-2). This step is crucial as it removes the logarithm, allowing us to solve for x directly. The value of e^(-2) is approximately 0.1353, but we will keep it in exponential form for now to avoid rounding errors.
4. Solve for x
To solve for x, we add 1 to both sides of the equation: x - 1 + 1 = e^(-2) + 1. This simplifies to: x = e^(-2) + 1. Now we have x isolated on one side of the equation. To find the numerical value of x, we evaluate e^(-2) + 1. Since we are asked to round the answer to the nearest hundredth, we will calculate the value of e^(-2) and then add 1.
5. Calculate the Value of x
The value of e^(-2) is approximately 0.135335. Adding 1 to this value gives us: x ≈ 0.135335 + 1 = 1.135335. Rounding this to the nearest hundredth, we get: x ≈ 1.14. This is the final solution for x.
Verification
It is essential to verify that our solution is valid by plugging it back into the original equation and ensuring that it satisfies the equation. We found that x ≈ 1.14. Plugging this value into the original equation, we get: 4 + ln(1.14 - 1) - 2 = 4 + ln(0.14) - 2. The natural logarithm of 0.14 is approximately -1.9661. Thus, the equation becomes: 4 + (-1.9661) - 2 ≈ 0.0339, which is very close to 0. The slight discrepancy is due to rounding. Since x = 1.14 > 1, our solution is within the domain of the natural logarithm, making it a valid solution.
Final Answer
The solution to the equation 4 + ln(x-1) - 2 = 0, rounded to the nearest hundredth, is:
x ≈ 1.14
This comprehensive guide provides a clear and detailed solution to the given equation, emphasizing the importance of understanding the properties of logarithms and algebraic manipulations. By following these steps, you can confidently solve similar equations involving logarithms.
Conclusion
Solving logarithmic equations requires a strong understanding of logarithmic properties and algebraic manipulation techniques. In this article, we walked through the process of solving the equation 4 + ln(x-1) - 2 = 0 step by step. We started by simplifying the equation, isolating the logarithmic term, exponentiating both sides, and finally solving for x. We also verified our solution to ensure its validity. The final solution, rounded to the nearest hundredth, is x ≈ 1.14. Mastering these techniques will enhance your problem-solving skills in mathematics and related fields. Remember to always check the domain of logarithmic functions to ensure the validity of your solutions. Practice with similar problems to solidify your understanding and build confidence in solving logarithmic equations.
Further Practice
To further enhance your understanding, try solving these similar problems:
- Solve for x: 5 + ln(x+2) - 3 = 0
- Solve for x: 2ln(x-3) = 4
- Solve for x: 10 - ln(2x+1) = 7
By working through these problems, you will reinforce your skills in solving logarithmic equations and gain a deeper understanding of the concepts involved.
