Solving Logarithmic Equations Representing Log4(x+3) = Log2(2+x)

$$\log _4(x+3)=\log _2(2+x)$$

A. $y_1=\frac{\log (x+3)}{\log 4}, y_2=\frac{\log (2+x)}{\log 2}$ B. $y_1=\frac{\log x+3}{\log 4}, y_2=\frac{\log 2+x}{\log 2}$ C. $y_1=\log _4(x+3), y_2=\log _2(2)+x$ D. $y_1=\log _4 x+3, y_2=\log _2 2+x$

Logarithmic equations are a fundamental part of mathematics, appearing in various fields such as calculus, algebra, and even real-world applications like calculating pH levels or Richter scale measurements. The equation presented, $\log _4(x+3)=\log _2(2+x)$, requires a solid understanding of logarithmic properties to solve effectively. To tackle this, we need to delve into the logarithmic properties, especially the change of base formula, which is instrumental in simplifying and solving such equations. Understanding the domain of logarithmic functions is crucial, as the arguments of logarithms must be positive. This means that in our equation, both $x+3$ and $2+x$ must be greater than zero. This initial constraint helps us to narrow down potential solutions and avoid extraneous roots. We need to consider transformations of logarithmic functions and how these transformations affect the graph and the solutions of the equation. Recognizing the relationship between logarithmic and exponential forms is also crucial, as one can often rewrite a logarithmic equation in exponential form to simplify the solving process. Moreover, different logarithmic bases can make comparison and simplification challenging. Hence, the change of base formula is not just a tool, but a necessity in many cases. Understanding the implications of this formula allows us to rewrite logarithms in a common base, making comparisons and algebraic manipulations easier. By understanding these core concepts, we can break down the problem into manageable parts and approach the solution systematically.

To determine which system of equations correctly represents the given logarithmic equation, $\log _4(x+3)=\log _2(2+x)$, we must understand how a single equation can be expressed as a system of two equations. The core idea here is to represent each side of the original equation as a separate function. This approach allows us to analyze and graph each function independently, with the solutions to the original equation corresponding to the points where the graphs of the two functions intersect. To achieve this transformation, we introduce two new variables, typically $y_1$ and $y_2$, each representing one side of the original equation. This technique is particularly useful when dealing with complex equations, as it simplifies the visualization and analysis process. For the left-hand side, $\log _4(x+3)$, we set $y_1$ equal to this expression. For the right-hand side, $\log _2(2+x)$, we set $y_2$ equal to this expression. This establishes a system of two equations: $y_1 = \log _4(x+3)$ and $y_2 = \log _2(2+x)$. The solutions to this system will be the values of $x$ for which $y_1$ and $y_2$ are equal, which corresponds to the solutions of the original logarithmic equation. This transformation not only aids in solving the equation algebraically but also provides a visual representation that can enhance understanding. The change of base formula plays a pivotal role in expressing these logarithmic functions in a form that is easier to manipulate and compare. This formula allows us to convert logarithms from one base to another, which is essential when dealing with different bases like 4 and 2 in our equation. This method allows us to verify the given options systematically and identify the one that correctly represents the original equation as a system of equations.

Now, let's analyze each of the provided options to determine which one correctly represents the given logarithmic equation, $\log _4(x+3)=\log _2(2+x)$, as a system of equations. We'll pay close attention to how each option expresses the logarithmic terms and compare them to the original equation.

Option A presents the system: $y_1=\frac{\log (x+3)}{\log 4}, y_2=\frac{\log (2+x)}{\log 2}$. This option utilizes the change of base formula correctly. The change of base formula states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ can be any base (commonly 10 or $e$). Applying this to our original equation, we can rewrite $\log _4(x+3)$ as $\frac{\log (x+3)}{\log 4}$ and $\log _2(2+x)$ as $\frac{\log (2+x)}{\log 2}$. Thus, option A accurately represents the original equation as a system of equations using the change of base formula. This accurate application of the change of base formula is crucial for solving logarithmic equations with different bases. The use of common logarithms (base 10) allows for easier computation and comparison, especially when using calculators or computational tools.

Option B: $y_1=\frac{\log x+3}{\log 4}, y_2=\frac{\log 2+x}{\log 2}$. This option incorrectly interprets the logarithmic terms. The expression $\frac{\log x+3}{\log 4}$ is not equivalent to $\log _4(x+3)$. Similarly, $\frac{\log 2+x}{\log 2}$ is not equivalent to $\log _2(2+x)$. This option seems to misunderstand the order of operations and the properties of logarithms, making it an incorrect representation of the original equation.

Option C: $y_1=\log _4(x+3), y_2=\log _2(2)+x$. This option correctly represents the left-hand side of the original equation, $y_1=\log _4(x+3)$, but incorrectly represents the right-hand side. The term $\log _2(2+x)$ is not the same as $\log _2(2)+x$. This option appears to incorrectly separate the logarithm and the argument, leading to a misrepresentation of the original equation.

Option D: $y_1=\log _4 x+3, y_2=\log _2 2+x$. This option incorrectly represents both sides of the original equation. The term $\log _4(x+3)$ is not the same as $\log _4 x+3$, and $\log _2(2+x)$ is not the same as $\log _2 2+x$. This option demonstrates a misunderstanding of how logarithmic functions operate and the importance of maintaining the correct order of operations and logarithmic properties.

Based on our analysis, Option A, $y_1=\frac{\log (x+3)}{\log 4}, y_2=\frac{\log (2+x)}{\log 2}$, is the only option that correctly represents the original logarithmic equation as a system of equations. This option accurately applies the change of base formula, which is essential for solving equations involving logarithms with different bases. The other options demonstrate misunderstandings of logarithmic properties and the correct application of mathematical transformations. Understanding the change of base formula, the properties of logarithms, and the process of transforming equations into systems are crucial skills in solving logarithmic equations effectively. By correctly applying these concepts, we can break down complex problems into manageable steps and arrive at accurate solutions.