Solving $e^{\ln (2 X+3)}$ A Step-by-Step Mathematical Guide

The question presented, $e^{\ln (2 x+3)}$, falls under the domain of exponential and logarithmic functions, a cornerstone of mathematics. To solve this, we must delve into the fundamental relationship between these two operations. Exponential and logarithmic functions, with the same base, are inverses of each other. This inverse relationship is the key to simplifying and solving expressions like the one given. In this article, we will not only dissect the problem step-by-step to arrive at the correct answer but also explore the underlying concepts and properties that govern these functions. Understanding these principles is crucial for tackling more complex problems in calculus, algebra, and various scientific fields. Let's begin by revisiting the core definition of logarithms and exponentials, and how they interact with each other. This foundation will provide a solid ground for comprehending the solution and its implications in a broader mathematical context.

To truly understand the expression $e^{\ln (2 x+3)}$, we must first grasp the inverse relationship between exponential and logarithmic functions. The logarithm, in simple terms, answers the question: "To what power must we raise the base to obtain a certain number?" The exponential function, on the other hand, performs the reverse operation. The expression $e^{\ln (2 x+3)}$ involves the natural logarithm, denoted as $\ln$, which has the base $e$ (Euler's number, approximately 2.71828). When we see $\ln(x)$, it means "the power to which we must raise $e$ to get $x$." Therefore, $e^{\ln(x)}$ essentially asks, "What do we get when we raise $e$ to the power that gives us $x$?" The fundamental property that governs this interaction is $e^{\ln(x)} = x$, provided that $x$ is positive. This identity is crucial because it simplifies the expression dramatically. It tells us that the exponential function with base $e$ and the natural logarithm cancel each other out, leaving us with the argument of the logarithm. This relationship is not just a mathematical trick; it's a fundamental characteristic of inverse functions. Understanding this principle allows us to tackle a wide range of problems involving exponential and logarithmic expressions. In the following sections, we will apply this principle to solve the given problem and explore some nuances of this relationship.

Now, let's apply our understanding of the inverse relationship between exponential and logarithmic functions to the given expression, $e^{\ln (2 x+3)}$. Recall the fundamental property we discussed: $e^{\ln(x)} = x$. This identity is the key to simplifying the expression. In our case, the argument of the natural logarithm is $(2x + 3)$. Therefore, we can directly apply the property, substituting $(2x + 3)$ for $x$ in the identity. This gives us $e^{\ln (2 x+3)} = 2x + 3$. This is a straightforward application of the inverse relationship. The exponential function with base $e$ effectively cancels out the natural logarithm, leaving us with the argument of the logarithm. However, there's an important caveat to consider: the domain of the natural logarithm. The natural logarithm is only defined for positive arguments. This means that $(2x + 3)$ must be greater than zero. We can express this condition mathematically as $2x + 3 > 0$. Solving this inequality gives us $x > -\frac{3}{2}$. This condition is crucial because it defines the values of $x$ for which our solution is valid. If $x$ is less than or equal to $-\frac{3}{2}$, the expression $\ln(2x + 3)$ is undefined, and consequently, the entire expression $e^{\ln (2 x+3)}$ is undefined. Therefore, the solution $2x + 3$ is only valid when $x > -\frac{3}{2}$. In the next section, we will discuss the implications of this domain restriction and its importance in the context of the problem.

As we've established, the solution to $e^{\ln (2 x+3)}$ is $2x + 3$, but this solution is contingent upon a critical condition: the argument of the natural logarithm, $(2x + 3)$, must be positive. This stems from the fundamental definition of logarithms, which are only defined for positive inputs. The logarithm of a non-positive number is undefined within the realm of real numbers. Mathematically, this constraint is expressed as $2x + 3 > 0$. Solving this inequality provides the domain of the function: $x > -\frac{3}{2}$. This domain restriction is not merely a technicality; it's a crucial aspect of the problem. It dictates the set of values for which our solution is valid. If we were to substitute a value of $x$ less than or equal to $-\frac{3}{2}$ into the original expression, we would encounter the logarithm of a non-positive number, rendering the expression undefined. For instance, if we let $x = -2$, then $2x + 3 = -1$, and $\ln(-1)$ is undefined in real numbers. Therefore, while the algebraic manipulation yields $2x + 3$, we must always remember the underlying domain restriction. This highlights the importance of considering domain restrictions when dealing with logarithmic and exponential functions. Failing to do so can lead to incorrect or nonsensical results. In practical applications, domain restrictions often represent physical or contextual limitations on the variables involved. In the following section, we will summarize the solution and emphasize the key takeaways from this problem.

In conclusion, the expression $e^{\ln (2 x+3)}$ simplifies to $2x + 3$ due to the inverse relationship between exponential and natural logarithmic functions. This simplification is a direct application of the identity $e^{\ln(x)} = x$. However, it's crucial to remember the domain restriction imposed by the natural logarithm. The argument of the logarithm, $(2x + 3)$, must be positive, leading to the condition $x > -\frac{3}{2}$. This domain restriction ensures the validity of the solution. The problem highlights the importance of understanding the fundamental properties of exponential and logarithmic functions, as well as the significance of considering domain restrictions. When working with these functions, it's not enough to simply perform algebraic manipulations; we must also ensure that the results are consistent with the definitions and limitations of the functions involved. This problem serves as a valuable illustration of these principles, providing a foundation for tackling more complex problems in mathematics involving exponential and logarithmic functions. By understanding the inverse relationship and the domain considerations, we can confidently navigate and solve a wide range of problems in calculus, algebra, and other scientific disciplines. The key takeaway is that mathematical rigor requires not only algebraic proficiency but also a deep understanding of the underlying concepts and constraints.

Final Answer: The final answer is $\boxed{2x+3}$