Solving The Logarithmic Equation 2ln(e^(ln(2x))) - Ln(e^(ln(10x))) = Ln(30)

$$2 \ln e^{\ln 2 x}-\ln e^{\ln 10 x}=\ln 30$$

Introduction: Decoding the Logarithmic Puzzle

In this mathematical journey, we aim to unravel the true solution to the logarithmic equation presented. Logarithmic equations, often perceived as intricate puzzles, require a systematic approach and a keen understanding of logarithmic properties. This article serves as a comprehensive guide, meticulously dissecting the equation, employing fundamental logarithmic principles, and navigating through the solution process. Our primary focus lies in transforming the equation into a manageable form, isolating the variable x, and ultimately determining the correct answer from the given options. Before diving into the specifics, let's emphasize the importance of logarithmic equations in various scientific and engineering fields. From modeling population growth to calculating compound interest, logarithms play a pivotal role. Mastering these equations is not just an academic exercise; it's a valuable skill applicable to real-world scenarios. So, let's embark on this exploration, demystify the logarithmic equation, and arrive at the precise solution.

Dissecting the Equation: A Step-by-Step Approach

To effectively solve the equation, we must first dissect it, understand its components, and identify the underlying logarithmic properties that can be applied. The given equation is:

$$2 \ln e^{\ln 2 x}-\ln e^{\ln 10 x}=\ln 30$$

At first glance, the equation may seem daunting, but a closer look reveals its inherent structure. We have logarithmic terms involving the natural logarithm (\ln) and exponential functions with base e. The key to simplifying this equation lies in understanding the inverse relationship between logarithms and exponentials. Specifically, we'll leverage the property that ${\ln e^y = y}$. This property allows us to eliminate the natural logarithm and exponential functions that are nested within each other. By carefully applying this property and other logarithmic rules, we can systematically reduce the complexity of the equation. The initial step involves simplifying the left-hand side of the equation by applying the inverse relationship between logarithms and exponentials. This will pave the way for further simplification and ultimately lead us to the solution. As we proceed, we'll emphasize each step, ensuring clarity and a thorough understanding of the underlying logic. This meticulous approach is crucial for solving not just this specific equation but any logarithmic equation you may encounter.

Simplifying the Left-Hand Side: Unveiling the Core

The core of our solution strategy lies in simplifying the left-hand side of the equation. We begin by applying the crucial property of logarithms: ${\ln e^y = y}$. This property states that the natural logarithm of an exponential function with base e simply equals the exponent. Applying this to our equation, we have:

$$2 \ln e^{\ln 2 x} = 2(\ln 2x)$$

and

$$\ln e^{\ln 10 x} = \ln 10x$$

Substituting these simplified terms back into the original equation, we get:

$$2(\ln 2x) - \ln 10x = \ln 30$$

Now, we've successfully eliminated the nested exponentials and logarithms, making the equation significantly more manageable. The next step involves further simplification using other logarithmic properties. Specifically, we'll employ the power rule of logarithms, which states that ${a \ln b = \ln b^a}$. This rule will allow us to bring the coefficient 2 inside the logarithm in the first term. By applying this rule, we continue to peel back the layers of complexity, bringing us closer to isolating the variable x. This step-by-step simplification is a testament to the power of logarithmic properties in solving complex equations. Remember, each step is a deliberate application of a fundamental rule, ensuring accuracy and a clear path towards the solution.

Leveraging Logarithmic Properties: Power and Product Rules

Having simplified the initial terms, we now delve deeper into the realm of logarithmic properties to further refine our equation. The next powerful tool in our arsenal is the power rule of logarithms, which states that ${a \ln b = \ln b^a}$. Applying this rule to the term ${2(\ln 2x)}$, we get:

$$2(\ln 2x) = \ln (2x)^2 = \ln (4x^2)$$

Substituting this back into our equation, we now have:

$$\ln (4x^2) - \ln (10x) = \ln 30$$

The equation is becoming increasingly streamlined. Next, we employ another fundamental logarithmic property: the quotient rule. The quotient rule states that ${\ln a - \ln b = \ln (a/b)}$. Applying this rule to the left-hand side of our equation, we get:

$$\ln (4x^2) - \ln (10x) = \ln \left(\frac{4x^2}{10x}\right) = \ln \left(\frac{2x}{5}\right)$$

Now our equation is significantly simplified:

$$\ln \left(\frac{2x}{5}\right) = \ln 30$$

The elegance of logarithmic properties is evident in this transformation. By strategically applying the power and quotient rules, we've condensed the equation into a concise form, ready for the final steps of solving for x. This process highlights the importance of mastering logarithmic properties – they are the keys to unlocking solutions to complex logarithmic equations.

Isolating the Variable: The Final Steps to the Solution

With our equation simplified to ${\ln \left(\frac{2x}{5}\right) = \ln 30}$, we are now on the verge of isolating the variable x. The beauty of this simplified form is that we can directly equate the arguments of the logarithms. Since the natural logarithms on both sides are equal, their arguments must also be equal. This gives us:

$$\frac{2x}{5} = 30$$

This equation is now a simple linear equation, easily solvable for x. To isolate x, we first multiply both sides of the equation by 5:

$$2x = 30 \times 5 = 150$$

Then, we divide both sides by 2:

$$x = \frac{150}{2} = 75$$

Thus, we have arrived at the solution: x = 75. This elegant solution is a direct result of the systematic simplification and application of logarithmic properties throughout the process. It underscores the importance of a step-by-step approach when tackling logarithmic equations. The final step is to verify this solution against the original equation to ensure its validity. This verification process is crucial to confirm that our solution is not extraneous and indeed satisfies the given equation. The satisfaction of arriving at the correct solution is amplified by the rigorous process we've followed, reinforcing the power of mathematical principles in problem-solving.

Verification: Ensuring the Solution's Validity

Having obtained the solution x = 75, the crucial next step is verification. We must substitute this value back into the original equation to ensure it holds true. This process is paramount in mathematics, especially when dealing with logarithmic equations, as it helps identify any extraneous solutions that may arise due to the nature of logarithmic functions. Our original equation is:

$$2 \ln e^{\ln 2 x}-\ln e^{\ln 10 x}=\ln 30$$

Substituting x = 75, we get:

$$2 \ln e^{\ln (2 \times 75)}-\ln e^{\ln (10 \times 75)}=\ln 30$$

Simplifying the expressions inside the logarithms:

$$2 \ln e^{\ln 150}-\ln e^{\ln 750}=\ln 30$$

Applying the property ${\ln e^y = y}$:

$$2(\ln 150) - \ln 750 = \ln 30$$

Using the power rule of logarithms:

$$\ln (150^2) - \ln 750 = \ln 30$$

$$\ln 22500 - \ln 750 = \ln 30$$

Applying the quotient rule of logarithms:

$$\ln \left(\frac{22500}{750}\right) = \ln 30$$

$$\ln 30 = \ln 30$$

The equation holds true! This verification confirms that x = 75 is indeed the valid solution to the given logarithmic equation. The satisfaction derived from this verification is significant, as it solidifies our confidence in the solution and the systematic approach we employed. This step-by-step verification process is a cornerstone of mathematical rigor, ensuring the accuracy and reliability of our results.

Conclusion: The Triumph of Logarithmic Mastery

In conclusion, we have successfully navigated the intricacies of the given logarithmic equation and arrived at the true solution: x = 75. This journey, from the initial equation to the final verification, has underscored the importance of understanding and applying fundamental logarithmic properties. We meticulously dissected the equation, employed the power rule, quotient rule, and the inverse relationship between logarithms and exponentials to simplify it step by step. The systematic approach not only led us to the solution but also reinforced the elegance and power of mathematical principles in problem-solving. The verification process further solidified our confidence in the solution, ensuring its validity and highlighting the importance of rigor in mathematical endeavors. The knowledge gained through this exercise extends beyond this specific equation. The skills and techniques we've honed are transferable to a wide range of logarithmic problems, empowering us to tackle future mathematical challenges with confidence. The mastery of logarithmic equations is not just an academic achievement; it's a valuable asset in various fields, enabling us to model and solve real-world problems effectively.

Final Answer

The correct answer is B. x=75.