Solving 5 + Ln(x-4) = 7 Step-by-Step Guide

In this article, we will delve into the step-by-step process of solving the equation 5 + 1n(x-4) = 7 for the variable x. This equation involves a natural logarithm, making it essential to understand logarithmic properties and how to manipulate them to isolate x. We'll walk through each step meticulously, providing clear explanations and rounding intermediate computations as instructed. By the end of this guide, you'll have a solid understanding of how to solve this type of equation and be able to apply the same principles to similar problems.

Understanding the Equation and Logarithms

Before we jump into the solution, let's first understand the components of the equation 5 + 1n(x-4) = 7. The key element here is the natural logarithm, denoted as "ln". A logarithm is essentially the inverse operation of exponentiation. In simpler terms, if we have an equation like b^y = x, the logarithm (base b) of x is y, written as log_b(x) = y. The natural logarithm is a special case where the base is the mathematical constant e, approximately equal to 2.71828. So, ln(x) is the same as log_e(x). Understanding this relationship between exponentiation and logarithms is crucial for solving equations involving logarithms.

Logarithmic properties are the foundation for manipulating and solving logarithmic equations. One of the most important properties is the inverse relationship between exponential and logarithmic functions. This means that e^(ln(x)) = x and ln(e^x) = x. We will use this property later in our solution. Another important property is the power rule of logarithms, which states that ln(a^b) = b * ln(a). Additionally, we have the product rule, ln(a*b) = ln(a) + ln(b), and the quotient rule, ln(a/b) = ln(a) - ln(b). These properties allow us to combine or separate logarithmic terms, which is often necessary to isolate the variable.

The term (x - 4) inside the logarithm is called the argument. A crucial thing to remember about logarithms is that the argument must be strictly greater than zero. This is because we cannot take the logarithm of a negative number or zero. This constraint will be important when we check our solution later to ensure that it is valid.

Now that we have a solid understanding of logarithms, let's move on to the steps involved in solving the equation.

Step-by-Step Solution

Our goal is to isolate x on one side of the equation. To do this, we will follow these steps:

1. Isolate the Logarithmic Term

The first step is to isolate the term containing the natural logarithm, which is 1n(x-4). We can achieve this by subtracting 5 from both sides of the equation:

5 + ln(x - 4) = 7

Subtract 5 from both sides:

ln(x - 4) = 7 - 5

ln(x - 4) = 2

2. Convert from Logarithmic Form to Exponential Form

Now that we have the logarithmic term isolated, we can convert the equation from logarithmic form to exponential form. Recall that ln(x - 4) = 2 is equivalent to saying that e raised to the power of 2 equals (x - 4). In other words:

e^2 = x - 4

This step is crucial because it eliminates the logarithm and allows us to work with a more straightforward algebraic equation.

3. Isolate x

Next, we need to isolate x on one side of the equation. We can do this by adding 4 to both sides:

e^2 = x - 4

Add 4 to both sides:

x = e^2 + 4

4. Calculate the Value of e^2

Now, we need to calculate the value of e^2. Using a calculator, we find that:

e^2 ≈ 7.389

5. Approximate the Solution and Round to the Nearest Hundredth

Substitute the value of e^2 back into the equation:

x ≈ 7.389 + 4

x ≈ 11.389

Finally, we round the result to the nearest hundredth, as instructed:

x ≈ 11.39

Therefore, the approximate solution for x is 11.39.

Checking the Solution

It is essential to check our solution to ensure that it is valid. Remember that the argument of a logarithm must be greater than zero. In our original equation, the argument is (x - 4). So, we need to verify that (11.39 - 4) > 0:

11. 39 - 4 = 7.39

Since 7.39 is greater than 0, our solution is valid.

Conclusion

In this comprehensive guide, we have demonstrated how to solve the equation 5 + ln(x - 4) = 7 for x. We meticulously followed each step, explaining the underlying principles of logarithms and their properties. By isolating the logarithmic term, converting to exponential form, and isolating x, we arrived at the solution x ≈ 11.39. We also emphasized the importance of checking the solution to ensure its validity. This step-by-step approach can be applied to solve a wide range of logarithmic equations. Remember to always consider the domain of the logarithmic function and verify that your solutions are within the valid range. Understanding the concepts and practicing regularly will make you confident in solving complex mathematical problems.

By understanding the core concepts and following a systematic approach, you can confidently tackle similar problems involving logarithmic equations. Practice is key to mastering these skills, so don't hesitate to work through additional examples. This comprehensive guide equips you with the knowledge and techniques necessary to solve for x in various logarithmic scenarios. Keep practicing and building your mathematical confidence!

This thorough exploration of solving 5 + ln(x-4) = 7 not only provides a step-by-step solution but also deepens understanding of logarithmic functions and their applications. By mastering these techniques, you enhance your problem-solving capabilities in mathematics and related fields. Remember, the journey of learning mathematics is a continuous process of building knowledge and refining skills. Keep exploring, keep practicing, and keep growing!