Solving Ln(x-2) - Ln(2) = -3 A Step-by-Step Guide

In this article, we will explore how to solve the equation ln(x-2) - ln(2) = -3. This equation involves logarithmic functions, and to find the value of x, we need to understand the properties of logarithms and how to manipulate them. This comprehensive guide will walk you through each step, ensuring a clear understanding of the process. Whether you're a student tackling algebra problems or someone looking to refresh your math skills, this explanation will provide the knowledge and confidence you need.

Understanding Logarithmic Equations

Before diving into the solution, it’s crucial to grasp the fundamentals of logarithmic equations. Logarithmic functions are the inverse of exponential functions. The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is an irrational number approximately equal to 2.71828. A logarithmic equation is an equation that involves logarithms of expressions containing a variable. Solving these equations often requires using the properties of logarithms to simplify and isolate the variable.

Key properties of logarithms that will be useful in solving our equation include:

1. The quotient rule: ln(a) - ln(b) = ln(a/b)
2. The exponential form: If ln(a) = b, then a = e^b

These properties allow us to combine logarithmic terms and convert logarithmic equations into exponential forms, making them easier to solve. Understanding these concepts is the first step in tackling more complex logarithmic problems.

Step 1: Combining Logarithmic Terms

The first step in solving the equation ln(x-2) - ln(2) = -3 is to combine the logarithmic terms on the left side. We can use the quotient rule of logarithms, which states that ln(a) - ln(b) = ln(a/b). Applying this rule, we combine the two logarithms into a single term:

ln((x-2)/2) = -3

This step simplifies the equation by reducing the number of logarithmic terms, making it easier to work with. By combining the terms, we are essentially condensing the expression into a more manageable form. This is a crucial step because it allows us to move towards isolating the variable x. Recognizing and applying the appropriate logarithmic property is key to simplifying and solving these types of equations.

Step 2: Converting to Exponential Form

Now that we have a single logarithmic term, the next step is to convert the equation from logarithmic form to exponential form. Recall that the natural logarithm is the logarithm to the base e. So, if ln(a) = b, then a = e^b. Applying this to our equation, ln((x-2)/2) = -3, we can rewrite it in exponential form:

(x-2)/2 = e^(-3)

This conversion is a pivotal step because it eliminates the logarithm, allowing us to work directly with the algebraic expression. The exponential form provides a clearer path to isolate the variable x. Understanding how to transition between logarithmic and exponential forms is essential for solving logarithmic equations effectively. This step lays the groundwork for the subsequent steps where we will isolate x and find its value.

Step 3: Isolating the Variable x

With the equation in exponential form, (x-2)/2 = e^(-3), our next goal is to isolate the variable x. We can do this by performing a series of algebraic manipulations. First, we multiply both sides of the equation by 2 to eliminate the denominator:

x - 2 = 2e^(-3)

Next, we add 2 to both sides of the equation to isolate x:

x = 2e^(-3) + 2

This step is crucial because it brings us closer to the solution by getting x by itself on one side of the equation. Each operation we perform is aimed at simplifying the equation and removing any terms or coefficients that are preventing us from finding the value of x. By isolating the variable, we set the stage for the final step where we will approximate the value of x.

Step 4: Approximating the Solution

Now that we have isolated x, we have the equation x = 2e^(-3) + 2. To find an approximate numerical value for x, we need to evaluate the expression 2e^(-3) + 2. Using a calculator, we can find that e^(-3) is approximately 0.04979. Therefore:

x ≈ 2(0.04979) + 2 x ≈ 0.09958 + 2 x ≈ 2.09958

Rounding to a reasonable number of decimal places, we get:

x ≈ 2.10

This step provides us with a practical solution by converting the exact form into a numerical approximation. The approximation allows us to understand the magnitude of x and provides a concrete answer to the equation. Using a calculator to evaluate exponential and logarithmic expressions is a common practice in solving these types of problems. The final approximation gives us a clear and usable value for x.

Verification of the Solution

It’s always a good practice to verify our solution to ensure it is correct and does not introduce any errors. To verify our solution, we substitute x ≈ 2.10 back into the original equation, ln(x-2) - ln(2) = -3:

ln(2.10 - 2) - ln(2) = -3 ln(0.10) - ln(2) ≈ -3

Using a calculator:

ln(0.10) ≈ -2.30259 ln(2) ≈ 0.69315

So,

-2.30259 - 0.69315 ≈ -2.99574

Which is approximately -3. Due to rounding, there is a slight difference, but the result is close enough to confirm that our solution is correct. This verification step is crucial because it helps to catch any potential errors made during the solving process. By substituting the value back into the original equation, we ensure that it satisfies the equation and is a valid solution.

Extraneous Solutions

In the context of logarithmic equations, it's essential to be aware of extraneous solutions. An extraneous solution is a value that satisfies the transformed equation but not the original equation. These solutions often arise due to the domain restrictions of logarithmic functions. Specifically, the argument of a logarithm must be greater than zero. In our original equation, ln(x-2) - ln(2) = -3, the term ln(x-2) is only defined when x-2 > 0, which means x > 2.

Our solution, x ≈ 2.10, satisfies this condition. If we had obtained a solution less than or equal to 2, it would be an extraneous solution and we would discard it. Checking for extraneous solutions is a critical step in solving logarithmic equations to ensure the validity of the answers. This involves confirming that the solution does not violate any domain restrictions of the original logarithmic expressions.

Conclusion

In this comprehensive guide, we have successfully solved the equation ln(x-2) - ln(2) = -3. We began by understanding the properties of logarithms, then combined logarithmic terms using the quotient rule, converted the equation to exponential form, isolated the variable x, and finally approximated the solution. We also emphasized the importance of verifying the solution and checking for extraneous solutions.

By following these steps, you can confidently tackle similar logarithmic equations. Remember to always apply the properties of logarithms correctly and be mindful of the domain restrictions. With practice, solving logarithmic equations will become a straightforward and manageable task. This methodical approach ensures accuracy and a deep understanding of the underlying mathematical principles.