Solving Logarithmic Equations Ln(x+3) - Ln(4) = 4

Understanding Logarithmic Equations

When it comes to solving logarithmic equations, a solid grasp of logarithmic properties is crucial. These properties act as the foundational tools that allow us to manipulate and simplify complex equations, making them more manageable and ultimately leading us to the solution. In this article, we are presented with the equation ln(x+3) - ln(4) = 4, which involves natural logarithms. To successfully solve for x, we'll need to delve into the fundamental logarithmic properties, particularly the quotient rule of logarithms. This rule states that the logarithm of a quotient is equal to the difference of the logarithms. Mathematically, it can be expressed as ln(a) - ln(b) = ln(a/b). This property is the key to condensing the left-hand side of our equation, paving the way for us to isolate x. Beyond the quotient rule, understanding the relationship between logarithms and exponential functions is paramount. The logarithmic function is essentially the inverse of the exponential function. This means that if we have an equation in logarithmic form, we can convert it to exponential form and vice versa. Specifically, for natural logarithms (ln), the base is the mathematical constant e, approximately equal to 2.71828. The relationship can be expressed as ln(a) = b is equivalent to e^b = a. This conversion is often the critical step in eliminating the logarithm and bringing the variable x into the open, where we can finally solve for it. The power of logarithms lies in their ability to transform multiplication and division problems into simpler addition and subtraction problems, and exponentiation problems into multiplication problems. This transformation is incredibly useful in various fields, including mathematics, physics, engineering, and computer science. Logarithms are used to model phenomena that span many orders of magnitude, such as the Richter scale for earthquake intensity, the pH scale for acidity, and the decibel scale for sound intensity. By mastering the fundamental properties of logarithms and their relationship with exponential functions, you unlock a powerful tool for solving a wide range of problems. In the following sections, we will apply these principles to solve our specific equation, ln(x+3) - ln(4) = 4, step by step, illustrating the practical application of these concepts.

Step-by-Step Solution

Let's begin by tackling the equation ln(x+3) - ln(4) = 4. Our primary goal is to isolate x, but first, we need to simplify the left-hand side of the equation. This is where the quotient rule of logarithms comes into play. As mentioned earlier, this rule states that ln(a) - ln(b) = ln(a/b). Applying this rule to our equation, we can combine the two logarithmic terms into a single logarithm: ln((x+3)/4) = 4. Now, we have a more condensed equation, but x is still trapped inside the logarithm. To free x, we need to convert the equation from logarithmic form to exponential form. Recall that ln(a) = b is equivalent to e^b = a. In our case, this means that ln((x+3)/4) = 4 is equivalent to e^4 = (x+3)/4. We have successfully eliminated the logarithm and now have a straightforward algebraic equation to solve. The next step is to isolate the term (x+3). We can achieve this by multiplying both sides of the equation by 4: 4 * e^4 = x + 3. Now, x is just one step away from being completely isolated. To get x by itself, we simply subtract 3 from both sides of the equation: x = 4 * e^4 - 3. At this point, we have an exact solution for x, but it's in terms of e, which is an irrational number. To get a numerical approximation, we need to use a calculator. Using a calculator, we find that e^4 is approximately 54.598. Therefore, 4 * e^4 is approximately 218.392. Finally, subtracting 3 from this result gives us x ≈ 218.392 - 3 = 215.392. The problem asks us to round our answer to the nearest hundredth. So, rounding 215.392 to the nearest hundredth, we get x ≈ 215.39. Thus, the solution to the equation ln(x+3) - ln(4) = 4, rounded to the nearest hundredth, is 215.39.

Verification and Conclusion

To ensure the accuracy of our solution, it's always a good practice to verify the answer. This involves substituting our calculated value of x back into the original equation and checking if both sides of the equation are equal. Our calculated solution is x ≈ 215.39. Substituting this value into the original equation, ln(x+3) - ln(4) = 4, we get: ln(215.39 + 3) - ln(4) = ln(218.39) - ln(4). Using a calculator, we find that ln(218.39) ≈ 5.386 and ln(4) ≈ 1.386. Therefore, ln(218.39) - ln(4) ≈ 5.386 - 1.386 = 4. This confirms that our solution, x ≈ 215.39, is indeed correct. In conclusion, we have successfully solved the logarithmic equation ln(x+3) - ln(4) = 4 by applying the quotient rule of logarithms to simplify the equation, converting it to exponential form to isolate x, and then solving the resulting algebraic equation. We obtained the solution x = 4 * e^4 - 3, which, when rounded to the nearest hundredth, gives us x ≈ 215.39. We then verified our solution by substituting it back into the original equation and confirming its validity. This problem highlights the importance of understanding and applying logarithmic properties, as well as the relationship between logarithmic and exponential functions, in solving logarithmic equations. The ability to manipulate logarithmic expressions and convert between logarithmic and exponential forms is a valuable skill in mathematics and its applications.

Therefore, the final answer is:

x = 215.39