Which Equation Has $x = -6$ As The Solution?A. Log ⁡ X 36 = 2 \log_x 36 = 2 Lo G X ​ 36 = 2 B. Log ⁡ 3 ( 2 X − 9 ) = 3 \log_3 (2x - 9) = 3 Lo G 3 ​ ( 2 X − 9 ) = 3 C. Log ⁡ 3 216 = X \log_3 216 = X Lo G 3 ​ 216 = X D. Log ⁡ 3 ( − 2 X − 3 ) = 2 \log_3 (-2x - 3) = 2 Lo G 3 ​ ( − 2 X − 3 ) = 2

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will explore the process of solving logarithmic equations and apply it to a specific problem. We will examine four different equations and determine which one has a solution of $x = -6$.

What are Logarithmic Equations?

A logarithmic equation is an equation that involves a logarithmic function. The logarithmic function is the inverse of the exponential function, and it is used to solve equations that involve exponential expressions. Logarithmic equations can be written in the form $\log_a b = c$, where $a$ is the base of the logarithm, $b$ is the argument of the logarithm, and $c$ is the result of the logarithm.

Properties of Logarithms

Before we can solve logarithmic equations, we need to understand the properties of logarithms. The three main properties of logarithms are:

• Product Property: $\log_a (b \cdot c) = \log_a b + \log_a c$
• Quotient Property: $\log_a (b / c) = \log_a b - \log_a c$
• Power Property: $\log_a (b^c) = c \cdot \log_a b$

Solving Logarithmic Equations

To solve a logarithmic equation, we need to isolate the logarithmic expression. We can do this by using the properties of logarithms to simplify the equation. Once we have isolated the logarithmic expression, we can use the definition of a logarithm to solve for the variable.

Equation A: $\log_x 36 = 2$

Let's start by examining Equation A: $\log_x 36 = 2$. To solve this equation, we need to isolate the logarithmic expression. We can do this by using the definition of a logarithm:

$$\log_x 36 = 2 \Rightarrow x^2 = 36 \Rightarrow x = \pm 6$$

Since we are looking for a solution of $x = -6$, we can see that Equation A does not have a solution of $x = -6$.

Equation B: $\log_3 (2x - 9) = 3$

Next, let's examine Equation B: $\log_3 (2x - 9) = 3$. To solve this equation, we need to isolate the logarithmic expression. We can do this by using the definition of a logarithm:

$$\log_3 (2x - 9) = 3 \Rightarrow 2x - 9 = 3^3 \Rightarrow 2x - 9 = 27 \Rightarrow 2x = 36 \Rightarrow x = 18$$

Since we are looking for a solution of $x = -6$, we can see that Equation B does not have a solution of $x = -6$.

Equation C: $\log_3 216 = x$

Now, let's examine Equation C: $\log_3 216 = x$. To solve this equation, we need to isolate the logarithmic expression. We can do this by using the definition of a logarithm:

$$\log_3 216 = x \Rightarrow 3^x = 216 \Rightarrow 3^x = 6^3 \Rightarrow 3^x = (3^2)^3 \Rightarrow 3^x = 3^6 \Rightarrow x = 6$$

Since we are looking for a solution of $x = -6$, we can see that Equation C does not have a solution of $x = -6$.

Equation D: $\log_3 (-2x - 3) = 2$

Finally, let's examine Equation D: $\log_3 (-2x - 3) = 2$. To solve this equation, we need to isolate the logarithmic expression. We can do this by using the definition of a logarithm:

$$\log_3 (-2x - 3) = 2 \Rightarrow -2x - 3 = 3^2 \Rightarrow -2x - 3 = 9 \Rightarrow -2x = 12 \Rightarrow x = -6$$

Since we are looking for a solution of $x = -6$, we can see that Equation D has a solution of $x = -6$.

Conclusion

In this article, we have examined four different logarithmic equations and determined which one has a solution of $x = -6$. We have seen that Equation A, Equation B, and Equation C do not have a solution of $x = -6$, while Equation D does have a solution of $x = -6$. By understanding the properties of logarithms and using the definition of a logarithm, we can solve logarithmic equations and determine which ones have a specific solution.

Key Takeaways

• Logarithmic equations involve a logarithmic function and can be written in the form $\log_a b = c$.
• The three main properties of logarithms are the product property, the quotient property, and the power property.
• To solve a logarithmic equation, we need to isolate the logarithmic expression and use the definition of a logarithm to solve for the variable.
• Equation D: $\log_3 (-2x - 3) = 2$ has a solution of $x = -6$.

Further Reading

If you are interested in learning more about logarithmic equations, I recommend checking out the following resources:

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

Introduction

In our previous article, we explored the concept of logarithmic equations and solved four different equations to determine which one has a solution of $x = -6$. In this article, we will answer some frequently asked questions about logarithmic equations and provide additional examples to help you understand the concept better.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function. The logarithmic function is the inverse of the exponential function, and it is used to solve equations that involve exponential expressions.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic expression and use the definition of a logarithm to solve for the variable. You can use the properties of logarithms, such as the product property, the quotient property, and the power property, to simplify the equation.

Q: What are the properties of logarithms?

A: The three main properties of logarithms are:

• Product Property: $\log_a (b \cdot c) = \log_a b + \log_a c$
• Quotient Property: $\log_a (b / c) = \log_a b - \log_a c$
• Power Property: $\log_a (b^c) = c \cdot \log_a b$

Q: How do I use the properties of logarithms to simplify an equation?

A: To use the properties of logarithms to simplify an equation, you need to identify the logarithmic expression and apply the appropriate property. For example, if you have the equation $\log_a (b \cdot c)$, you can use the product property to simplify it to $\log_a b + \log_a c$.

Q: What is the definition of a logarithm?

A: The definition of a logarithm is:

$$\log_a b = c \Rightarrow a^c = b$$

Q: How do I use the definition of a logarithm to solve for the variable?

A: To use the definition of a logarithm to solve for the variable, you need to isolate the logarithmic expression and apply the definition. For example, if you have the equation $\log_a b = c$, you can use the definition to rewrite it as $a^c = b$.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

• Not isolating the logarithmic expression: Make sure to isolate the logarithmic expression before applying the definition of a logarithm.
• Not using the correct property of logarithms: Make sure to use the correct property of logarithms to simplify the equation.
• Not checking the domain of the logarithmic function: Make sure to check the domain of the logarithmic function to ensure that the solution is valid.

Q: What are some real-world applications of logarithmic equations?

A: Logarithmic equations have many real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design and optimize systems.

Q: How can I practice solving logarithmic equations?

A: You can practice solving logarithmic equations by:

• Working through examples: Work through examples of logarithmic equations to practice solving them.
• Using online resources: Use online resources, such as Khan Academy and Mathway, to practice solving logarithmic equations.
• Taking a course: Take a course on logarithmic equations to learn more about the concept and practice solving them.

Conclusion

In this article, we have answered some frequently asked questions about logarithmic equations and provided additional examples to help you understand the concept better. We have also discussed some common mistakes to avoid when solving logarithmic equations and provided some real-world applications of logarithmic equations. By practicing solving logarithmic equations, you can become more confident and proficient in solving them.