Which Is The Only Solution To The Equation $\log_3(x^2+6x)=\log_3(2x+12$\]?A. $x = -6$ B. $x = -2$ C. $x = 0$ D. $x = 2$ E. $x = 6$

Introduction

Logarithmic equations can be challenging to solve, especially when they involve variables within the logarithm. In this article, we will focus on solving the equation $\log_3(x^2+6x)=\log_3(2x+12)$, which is a classic example of a logarithmic equation. We will break down the solution into manageable steps, using algebraic manipulations and properties of logarithms to find the value of $x$ that satisfies the equation.

Understanding the Equation

The given equation is $\log_3(x^2+6x)=\log_3(2x+12)$. To solve this equation, we need to understand the properties of logarithms. Specifically, we will use the property that states $\log_a(b) = \log_a(c)$ if and only if $b = c$. This property allows us to equate the expressions inside the logarithms.

Equating the Expressions Inside the Logarithms

Using the property mentioned above, we can equate the expressions inside the logarithms:

$$x^2+6x=2x+12$$

Simplifying the Equation

Now, we can simplify the equation by combining like terms:

$$x^2+4x-12=0$$

Factoring the Quadratic Equation

The quadratic equation $x^2+4x-12=0$ can be factored as:

$$(x+6)(x-2)=0$$

Solving for $x$

To solve for $x$, we can set each factor equal to zero and solve for $x$:

$$x+6=0 \Rightarrow x=-6$$

$$x-2=0 \Rightarrow x=2$$

Checking the Solutions

Before we can conclude that $x=-6$ and $x=2$ are the solutions to the equation, we need to check if they satisfy the original equation. We can do this by plugging each value of $x$ back into the original equation and checking if the equation holds true.

Checking $x=-6$

Plugging $x=-6$ into the original equation, we get:

$$\log_3((-6)^2+6(-6))=\log_3(2(-6)+12)$$

$$\log_3(36-36)=\log_3(-12+12)$$

$$\log_3(0)=\log_3(0)$$

This equation is true, so $x=-6$ is a solution to the equation.

Checking $x=2$

Plugging $x=2$ into the original equation, we get:

$$\log_3((2)^2+6(2))=\log_3(2(2)+12)$$

$$\log_3(4+12)=\log_3(4+12)$$

$$\log_3(16)=\log_3(16)$$

This equation is also true, so $x=2$ is a solution to the equation.

Conclusion

In this article, we solved the logarithmic equation $\log_3(x^2+6x)=\log_3(2x+12)$ using algebraic manipulations and properties of logarithms. We found that the solutions to the equation are $x=-6$ and $x=2$. We also checked these solutions by plugging them back into the original equation and verifying that they satisfy the equation.

Final Answer

The final answer to the equation $\log_3(x^2+6x)=\log_3(2x+12)$ is:

• $x = -6$
• $x = 2$

These are the only solutions to the equation, and they can be verified by plugging them back into the original equation.

Additional Tips and Tricks

When solving logarithmic equations, it's essential to remember the properties of logarithms, such as the property that states $\log_a(b) = \log_a(c)$ if and only if $b = c$. This property allows us to equate the expressions inside the logarithms and simplify the equation.

Additionally, when factoring quadratic equations, it's crucial to remember that the factors of the quadratic expression must be equal to zero. This means that we can set each factor equal to zero and solve for $x$.

Finally, when checking the solutions, it's essential to plug the values of $x$ back into the original equation and verify that they satisfy the equation. This ensures that the solutions are correct and that they satisfy the original equation.

Common Mistakes to Avoid

When solving logarithmic equations, there are several common mistakes to avoid. These include:

• Not using the properties of logarithms correctly
• Not simplifying the equation correctly
• Not factoring the quadratic equation correctly
• Not checking the solutions correctly

By avoiding these common mistakes, you can ensure that you solve logarithmic equations correctly and accurately.

Conclusion

In conclusion, solving logarithmic equations requires a deep understanding of the properties of logarithms and algebraic manipulations. By using these properties and techniques, you can solve logarithmic equations accurately and efficiently. Remember to check your solutions carefully and avoid common mistakes to ensure that you get the correct answer.

Final Thoughts

Solving logarithmic equations is an essential skill in mathematics, and it has numerous applications in real-world problems. By mastering this skill, you can solve a wide range of problems and make informed decisions. Remember to practice regularly and seek help when needed to improve your skills and confidence.

References

• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Solving Logarithmic Equations" by Khan Academy
• [3] "Logarithmic Equations and Inequalities" by Paul's Online Math Notes

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources.

Introduction

Logarithmic equations can be challenging to solve, especially for those who are new to the concept. In this article, we will address some of the most frequently asked questions about logarithmic equations, providing clear and concise answers to help you better understand the topic.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, a logarithmic equation is an equation that involves a variable or expression that is raised to a power, and the result is equal to a given value.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to use the properties of logarithms, such as the property that states $\log_a(b) = \log_a(c)$ if and only if $b = c$. You also need to use algebraic manipulations, such as factoring and simplifying the equation.

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

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

• Not using the properties of logarithms correctly
• Not simplifying the equation correctly
• Not factoring the quadratic equation correctly
• Not checking the solutions correctly

Q: How do I check my solutions to a logarithmic equation?

A: To check your solutions to a logarithmic equation, you need to plug the values of $x$ back into the original equation and verify that they satisfy the equation. This ensures that the solutions are correct and that they satisfy the original equation.

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

A: Logarithmic equations have numerous 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 and exercises in a textbook or online resource. You can also try solving logarithmic equations on your own, using online tools and resources to check your work.

Q: What are some resources for learning more about logarithmic equations?

A: Some resources for learning more about logarithmic equations include:

• Textbooks: "Algebra and Trigonometry" by Michael Sullivan, "Calculus" by James Stewart
• Online resources: Khan Academy, Math Open Reference, Paul's Online Math Notes
• Video tutorials: 3Blue1Brown, Crash Course, Vi Hart

Conclusion

In conclusion, logarithmic equations are an essential part of mathematics, with numerous real-world applications. By understanding the properties of logarithms and using algebraic manipulations, you can solve logarithmic equations accurately and efficiently. Remember to check your solutions carefully and avoid common mistakes to ensure that you get the correct answer.

Final Thoughts

Solving logarithmic equations is an essential skill in mathematics, and it has numerous applications in real-world problems. By mastering this skill, you can solve a wide range of problems and make informed decisions. Remember to practice regularly and seek help when needed to improve your skills and confidence.

References

• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Solving Logarithmic Equations" by Khan Academy
• [3] "Logarithmic Equations and Inequalities" by Paul's Online Math Notes

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources.