Which Of The Following Shows The True Solution To The Logarithmic Equation Below? Log ⁡ ( X ) + Log ⁡ ( X + 5 ) = Log ⁡ ( 6 X + 12 \log(x) + \log(x+5) = \log(6x + 12 Lo G ( X ) + Lo G ( X + 5 ) = Lo G ( 6 X + 12 ]A. X = − 3 X = -3 X = − 3 B. X = 4 X = 4 X = 4 C. X = − 3 X = -3 X = − 3 And X = 4 X = 4 X = 4 D. X = − 3 X = -3 X = − 3 And X = − 4 X = -4 X = − 4

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore the solution to the logarithmic equation $\log(x) + \log(x+5) = \log(6x + 12)$. We will break down the solution step by step and provide a clear explanation of each step.

Understanding Logarithmic Properties

Before we dive into the solution, it's essential to understand the properties of logarithms. The logarithmic equation $\log(x) + \log(x+5) = \log(6x + 12)$ can be simplified using the property of logarithms that states $\log(a) + \log(b) = \log(ab)$. This property allows us to combine the two logarithmic terms on the left-hand side of the equation into a single logarithmic term.

Simplifying the Logarithmic Equation

Using the property of logarithms, we can simplify the equation as follows:

$$\log(x) + \log(x+5) = \log(6x + 12)$$

$$\log(x(x+5)) = \log(6x + 12)$$

$$x(x+5) = 6x + 12$$

Expanding and Simplifying the Equation

Now that we have simplified the logarithmic equation, we can expand and simplify it further. Expanding the left-hand side of the equation, we get:

$$x^2 + 5x = 6x + 12$$

Subtracting $6x$ from both sides of the equation, we get:

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

Factoring the Quadratic Equation

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

$$(x - 4)(x + 3) = 0$$

Solving for x

Now that we have factored the quadratic equation, we can solve for $x$ by setting each factor equal to zero:

$$x - 4 = 0 \Rightarrow x = 4$$

$$x + 3 = 0 \Rightarrow x = -3$$

Conclusion

In conclusion, the solution to the logarithmic equation $\log(x) + \log(x+5) = \log(6x + 12)$ is $x = 4$ and $x = -3$. These two values satisfy the equation and are the true solutions.

Answer

The correct answer is:

• B. $x = 4$
• C. $x = -3$ and $x = 4$

Discussion

The solution to the logarithmic equation $\log(x) + \log(x+5) = \log(6x + 12)$ is not as straightforward as it seems. The equation requires careful simplification and factoring to arrive at the correct solution. The use of logarithmic properties and quadratic equations is essential in solving this type of equation.

Tips and Tricks

When solving logarithmic equations, it's essential to remember the following tips and tricks:

• Use logarithmic properties to simplify the equation.
• Factor the quadratic equation to find the solutions.
• Check the solutions to ensure they satisfy the original equation.

By following these tips and tricks, you can solve logarithmic equations with ease and arrive at the correct solution.

Common Mistakes

When solving logarithmic equations, it's easy to make mistakes. Some common mistakes include:

• Failing to simplify the equation using logarithmic properties.
• Failing to factor the quadratic equation.
• Not checking the solutions to ensure they satisfy the original equation.

By avoiding these common mistakes, you can ensure that you arrive at the correct solution.

Real-World Applications

Logarithmic equations have many real-world applications. Some examples include:

• 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.

By understanding logarithmic equations and how to solve them, you can apply this knowledge to real-world problems and make informed decisions.

Conclusion

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will provide a Q&A guide to help you understand logarithmic equations and how to solve them.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function. Logarithmic functions are used to solve equations that involve exponential functions.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can use the property of logarithms that states $\log(a) + \log(b) = \log(ab)$. This property allows you to combine the two logarithmic terms on the left-hand side of the equation into a single logarithmic term.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you can use the factoring method. This involves finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation is an equation that involves a logarithmic function, while an exponential equation is an equation that involves an exponential function. Logarithmic equations are used to solve equations that involve exponential functions, while exponential equations are used to solve equations that involve logarithmic functions.

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

A: To check the solutions to a logarithmic equation, you can substitute each solution into the original equation and check if it is true. If the solution satisfies the original equation, then it is a valid solution.

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

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

• Failing to simplify the equation using logarithmic properties.
• Failing to factor the quadratic equation.
• Not checking the solutions to ensure they satisfy the original equation.

Q: How do I apply logarithmic equations to real-world problems?

A: Logarithmic equations have many real-world applications. Some examples include:

• 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: What are some tips and tricks for solving logarithmic equations?

A: Some tips and tricks for solving logarithmic equations include:

• Use logarithmic properties to simplify the equation.
• Factor the quadratic equation to find the solutions.
• Check the solutions to ensure they satisfy the original equation.

Q: How do I practice solving logarithmic equations?

A: To practice solving logarithmic equations, you can try solving problems on your own or using online resources such as worksheets and practice exams. You can also try solving real-world problems that involve logarithmic equations.

Conclusion

In conclusion, logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. By understanding logarithmic properties, factoring quadratic equations, and checking solutions, you can master the art of solving logarithmic equations and apply this knowledge to real-world problems. Remember to practice regularly and avoid common mistakes to ensure that you arrive at the correct solution.

Additional Resources

For additional resources on logarithmic equations, including worksheets, practice exams, and online tutorials, visit the following websites:

• Khan Academy: Khan Academy offers a comprehensive guide to logarithmic equations, including video tutorials and practice exercises.
• Mathway: Mathway is an online math problem solver that can help you solve logarithmic equations and other math problems.
• Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve logarithmic equations and other math problems.

By using these resources and practicing regularly, you can master the art of solving logarithmic equations and apply this knowledge to real-world problems.