Maria Wrote The Equation Log ⁡ ( X 2 ) + Log ⁡ ( 20 X 2 ) = Log ⁡ 8 \log \left(\frac{x}{2}\right) + \log \left(\frac{20}{x^2}\right) = \log 8 Lo G ( 2 X ​ ) + Lo G ( X 2 20 ​ ) = Lo G 8 . What Is The Solution To Maria's Equation?A. X = 3 10 X = \frac{3}{10} X = 10 3 ​ B. X = 4 5 X = \frac{4}{5} X = 5 4 ​ C. X = 5 4 X = \frac{5}{4} X = 4 5 ​ D.

Introduction

Maria's equation is a logarithmic equation that involves the sum of two logarithmic expressions. The equation is given by $\log \left(\frac{x}{2}\right) + \log \left(\frac{20}{x^2}\right) = \log 8$. In this article, we will solve Maria's equation step by step and find the solution to the equation.

Understanding Logarithmic Properties

Before we start solving Maria's equation, it's essential to understand some logarithmic properties. The logarithmic properties that we will use in this article are:

• Product Property: $\log (ab) = \log a + \log b$
• Quotient Property: $\log \left(\frac{a}{b}\right) = \log a - \log b$
• Power Property: $\log (a^b) = b \log a$

Step 1: Simplify the Equation

To simplify Maria's equation, we can use the product property of logarithms. The product property states that the logarithm of a product is equal to the sum of the logarithms of the factors. We can rewrite the equation as:

$\log \left(\frac{x}{2}\right) + \log \left(\frac{20}{x^2}\right) = \log 8$

Using the product property, we can rewrite the equation as:

$\log \left(\frac{x}{2} \cdot \frac{20}{x^2}\right) = \log 8$

Simplifying the expression inside the logarithm, we get:

$\log \left(\frac{20x}{2x^2}\right) = \log 8$

Step 2: Simplify the Expression Inside the Logarithm

To simplify the expression inside the logarithm, we can cancel out the common factors. We can rewrite the expression as:

$\log \left(\frac{10}{x}\right) = \log 8$

Step 3: Equate the Expressions Inside the Logarithms

Since the logarithms are equal, we can equate the expressions inside the logarithms. We get:

$\frac{10}{x} = 8$

Step 4: Solve for x

To solve for x, we can multiply both sides of the equation by x. We get:

$10 = 8x$

Dividing both sides of the equation by 8, we get:

$x = \frac{10}{8}$

Simplifying the fraction, we get:

$x = \frac{5}{4}$

Conclusion

In this article, we solved Maria's equation step by step and found the solution to the equation. The solution to Maria's equation is $x = \frac{5}{4}$. This solution satisfies the original equation and is the correct answer.

Answer

The correct answer is:

• C. $x = \frac{5}{4}$

Discussion

Maria's equation is a logarithmic equation that involves the sum of two logarithmic expressions. The equation is given by $\log \left(\frac{x}{2}\right) + \log \left(\frac{20}{x^2}\right) = \log 8$. In this article, we solved Maria's equation step by step and found the solution to the equation. The solution to Maria's equation is $x = \frac{5}{4}$. This solution satisfies the original equation and is the correct answer.

Related Topics

• Logarithmic Equations: Logarithmic equations are equations that involve logarithmic expressions. They can be solved using logarithmic properties and algebraic manipulations.
• Logarithmic Properties: Logarithmic properties are rules that govern the behavior of logarithmic expressions. They include the product property, quotient property, and power property.
• Algebraic Manipulations: Algebraic manipulations are techniques used to simplify and solve algebraic expressions. They include factoring, combining like terms, and canceling out common factors.
  Maria's Equation: A Q&A Guide ==============================

Introduction

Maria's equation is a logarithmic equation that involves the sum of two logarithmic expressions. The equation is given by $\log \left(\frac{x}{2}\right) + \log \left(\frac{20}{x^2}\right) = \log 8$. In this article, we will answer some frequently asked questions about Maria's equation and provide a step-by-step guide to solving the equation.

Q: What is Maria's equation?

A: Maria's equation is a logarithmic equation that involves the sum of two logarithmic expressions. The equation is given by $\log \left(\frac{x}{2}\right) + \log \left(\frac{20}{x^2}\right) = \log 8$.

Q: How do I solve Maria's equation?

A: To solve Maria's equation, you can use the product property of logarithms to simplify the equation. Then, you can equate the expressions inside the logarithms and solve for x.

Q: What are the steps to solve Maria's equation?

A: The steps to solve Maria's equation are:

1. Simplify the equation using the product property of logarithms.
2. Equate the expressions inside the logarithms.
3. Solve for x.

Q: What is the solution to Maria's equation?

A: The solution to Maria's equation is $x = \frac{5}{4}$.

Q: Why is the solution $x = \frac{5}{4}$?

A: The solution $x = \frac{5}{4}$ is the correct answer because it satisfies the original equation and is the result of the algebraic manipulations.

Q: What are some common mistakes to avoid when solving Maria's equation?

A: Some common mistakes to avoid when solving Maria's equation include:

• Not using the product property of logarithms to simplify the equation.
• Not equating the expressions inside the logarithms.
• Not solving for x correctly.

Q: How can I practice solving logarithmic equations like Maria's equation?

A: You can practice solving logarithmic equations like Maria's equation by:

• Working through example problems.
• Using online resources and practice tests.
• Seeking help from a teacher or tutor.

Conclusion

In this article, we answered some frequently asked questions about Maria's equation and provided a step-by-step guide to solving the equation. We also discussed some common mistakes to avoid and provided tips for practicing solving logarithmic equations. By following these steps and tips, you can become proficient in solving logarithmic equations like Maria's equation.

Related Topics

• Logarithmic Equations: Logarithmic equations are equations that involve logarithmic expressions. They can be solved using logarithmic properties and algebraic manipulations.
• Logarithmic Properties: Logarithmic properties are rules that govern the behavior of logarithmic expressions. They include the product property, quotient property, and power property.
• Algebraic Manipulations: Algebraic manipulations are techniques used to simplify and solve algebraic expressions. They include factoring, combining like terms, and canceling out common factors.

Additional Resources

• Logarithmic Equation Solver: A logarithmic equation solver is a tool that can help you solve logarithmic equations like Maria's equation.
• Logarithmic Properties Chart: A logarithmic properties chart is a chart that summarizes the logarithmic properties and can help you remember them.
• Algebraic Manipulation Practice: Algebraic manipulation practice is a way to practice and improve your skills in simplifying and solving algebraic expressions.