Which System Of Equations Could Be Graphed To Solve The Equation Below? Log ⁡ 0.5 X = Log ⁡ 3 ( 2 + X \log_{0.5} X = \log_3 (2 + X Lo G 0.5 ​ X = Lo G 3 ​ ( 2 + X ]A. Y 1 = Log ⁡ 0.5 Log ⁡ 3 , Y 2 = Log ⁡ X Log ⁡ ( 2 + X ) Y_1 = \frac{\log 0.5}{\log 3}, \quad Y_2 = \frac{\log X}{\log (2 + X)} Y 1 ​ = L O G 3 L O G 0.5 ​ , Y 2 ​ = L O G ( 2 + X ) L O G X ​ B. $y_1 = \frac{\log X}{\log 0.5}, \quad Y_2 =

Understanding the Problem

The given equation is $\log_{0.5} x = \log_3 (2 + x)$. This equation involves logarithms with different bases, and we are asked to determine which system of equations could be graphed to solve this equation. To approach this problem, we need to understand the properties of logarithms and how they can be manipulated to simplify the equation.

Properties of Logarithms

Logarithms have several important properties that can be used to simplify equations. One of the key properties is the change of base formula, which states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $a$, $b$, and $c$ are positive real numbers and $c \neq 1$. This formula allows us to change the base of a logarithm to a more convenient base.

Applying the Change of Base Formula

We can apply the change of base formula to the given equation to simplify it. Let's start by changing the base of the first logarithm to 10, which is a common base for logarithms. This gives us:

$$\log_{0.5} x = \frac{\log x}{\log 0.5}$$

Now, we can rewrite the equation as:

$$\frac{\log x}{\log 0.5} = \log_3 (2 + x)$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log_b a = \frac{\log_c a}{\log_c b}$. We can apply this property to the right-hand side of the equation to get:

$$\frac{\log x}{\log 0.5} = \frac{\log (2 + x)}{\log 3}$$

Equating the Expressions

Now, we can equate the two expressions on the left-hand side and the right-hand side of the equation. This gives us:

$$\frac{\log x}{\log 0.5} = \frac{\log (2 + x)}{\log 3}$$

Graphing the System of Equations

To graph the system of equations, we need to find the points of intersection between the two equations. We can do this by setting the two expressions equal to each other and solving for $x$. This gives us:

$$\frac{\log x}{\log 0.5} = \frac{\log (2 + x)}{\log 3}$$

Solving for x

To solve for $x$, we can start by cross-multiplying the two fractions. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a

Understanding the Problem

The given equation is $\log_{0.5} x = \log_3 (2 + x)$. This equation involves logarithms with different bases, and we are asked to determine which system of equations could be graphed to solve this equation. To approach this problem, we need to understand the properties of logarithms and how they can be manipulated to simplify the equation.

Properties of Logarithms

Logarithms have several important properties that can be used to simplify equations. One of the key properties is the change of base formula, which states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $a$, $b$, and $c$ are positive real numbers and $c \neq 1$. This formula allows us to change the base of a logarithm to a more convenient base.

Applying the Change of Base Formula

We can apply the change of base formula to the given equation to simplify it. Let's start by changing the base of the first logarithm to 10, which is a common base for logarithms. This gives us:

$$\log_{0.5} x = \frac{\log x}{\log 0.5}$$

Now, we can rewrite the equation as:

$$\frac{\log x}{\log 0.5} = \log_3 (2 + x)$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log_b a = \frac{\log_c a}{\log_c b}$. We can apply this property to the right-hand side of the equation to get:

$$\frac{\log x}{\log 0.5} = \frac{\log (2 + x)}{\log 3}$$

Equating the Expressions

Now, we can equate the two expressions on the left-hand side and the right-hand side of the equation. This gives us:

$$\frac{\log x}{\log 0.5} = \frac{\log (2 + x)}{\log 3}$$

Graphing the System of Equations

To graph the system of equations, we need to find the points of intersection between the two equations. We can do this by setting the two expressions equal to each other and solving for $x$. This gives us:

$$\frac{\log x}{\log 0.5} = \frac{\log (2 + x)}{\log 3}$$

Solving for x

To solve for $x$, we can start by cross-multiplying the two fractions. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a^b = b \log a$ to simplify the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Simplifying the Equation

To simplify the equation further, we can use the property of logarithms that states $\log a - \log b = \log \frac{a}{b}$. We can apply this property to the equation to get:

$$\log x \log 3 - \log (2 + x) \log 0.5 = 0$$

Solving for x

To solve for $x$, we can start by isolating the term with $x$ on one side of the equation. This gives us:

$$\log x \log 3 = \log (2 + x) \log 0.5$$

Using Properties of Logarithms

We can use the property of logarithms that states $\log a