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

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 and find the value of $x$ that satisfies the equation.

Understanding Logarithmic Equations

Before we dive into solving Maria's equation, let's briefly review the properties of logarithmic equations. A logarithmic equation is an equation that involves a logarithmic expression. The logarithmic expression is typically in the form $\log_b a$, where $b$ is the base of the logarithm and $a$ is the argument of the logarithm.

One of the key properties of logarithmic equations is the product rule, which states that $\log_b (m \cdot n) = \log_b m + \log_b n$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Another important property of logarithmic equations is the quotient rule, which states that $\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n$. This means that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.

Solving Maria's Equation

Now that we have reviewed the properties of logarithmic equations, let's apply these properties to solve Maria's equation.

The first step in solving Maria's equation is to combine the two logarithmic expressions on the left-hand side of the equation using the product rule. This gives us:

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

Simplifying the expression inside the logarithm, we get:

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

Now, we can simplify the expression inside the logarithm further by canceling out the common factors:

$$\log \left(\frac{20x}{2x^2}\right) = \log \left(\frac{10}{x}\right)$$

Now, we can equate the logarithmic expression to the right-hand side of the equation:

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

Since the logarithmic expressions are equal, we can drop the logarithms and equate the arguments:

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

Now, we can solve for $x$ by multiplying both sides of the equation by $x$:

$$10 = 8x$$

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

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

Simplifying the fraction, we get:

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

Therefore, the solution to Maria's equation is $x = \frac{5}{4}$.

Conclusion

In this article, we solved Maria's equation using the properties of logarithmic equations. We combined the two logarithmic expressions on the left-hand side of the equation using the product rule, simplified the expression inside the logarithm, and equated the logarithmic expression to the right-hand side of the equation. We then dropped the logarithms and equated the arguments, solved for $x$, and found that the solution to Maria's equation is $x = \frac{5}{4}$.

Discussion

The solution to Maria's equation is $x = \frac{5}{4}$. This means that the value of $x$ that satisfies the equation is $\frac{5}{4}$.

One possible answer choice is $x = \frac{4}{5}$. However, this is not the correct solution to Maria's equation. The correct solution is $x = \frac{5}{4}$.

Another possible answer choice is $x = \frac{3}{10}$. However, this is also not the correct solution to Maria's equation. The correct solution is $x = \frac{5}{4}$.

Therefore, the correct answer to Maria's equation is $x = \frac{5}{4}$.

References

• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Solving Logarithmic Equations" by Purplemath

Additional Resources

• [1] "Logarithmic Equations" by Khan Academy
• [2] "Solving Logarithmic Equations" by Mathway

FAQs

• Q: What is the solution to Maria's equation? A: The solution to Maria's equation is $x = \frac{5}{4}$.
• Q: What are the properties of logarithmic equations? A: The properties of logarithmic equations include the product rule and the quotient rule.
• Q: How do you solve logarithmic equations? A: To solve logarithmic equations, you can combine the logarithmic expressions using the product rule, simplify the expression inside the logarithm, and equate the logarithmic expression to the right-hand side of the equation.
  Maria's Equation: A Q&A Guide =====================================

Introduction

In our previous article, we solved Maria's equation and found the value of $x$ that satisfies the equation. In this article, we will provide a Q&A guide to help you understand the solution to Maria's equation and answer any questions you may have.

Q&A

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 you solve Maria's equation?

A: To solve Maria's equation, you can combine the two logarithmic expressions on the left-hand side of the equation using the product rule, simplify the expression inside the logarithm, and equate the logarithmic expression to the right-hand side of the equation.

Q: What are the properties of logarithmic equations?

A: The properties of logarithmic equations include the product rule and the quotient rule. The product rule states that $\log_b (m \cdot n) = \log_b m + \log_b n$, and the quotient rule states that $\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n$.

Q: How do you simplify the expression inside the logarithm?

A: To simplify the expression inside the logarithm, you can cancel out any common factors. In the case of Maria's equation, we can simplify the expression inside the logarithm by canceling out the common factors of $x$.

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

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

Q: Why is $x = \frac{5}{4}$ the solution to Maria's equation?

A: $x = \frac{5}{4}$ is the solution to Maria's equation because it satisfies the equation. When we substitute $x = \frac{5}{4}$ into the equation, we get $\log \left(\frac{5}{4}\right) + \log \left(\frac{20}{\left(\frac{5}{4}\right)^2}\right) = \log 8$, which is true.

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

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

• Not combining the logarithmic expressions using the product rule
• Not simplifying the expression inside the logarithm
• Not equating the logarithmic expression to the right-hand side of the equation
• Not checking the solution to make sure it satisfies the equation

Q: How can I practice solving logarithmic equations?

A: You can practice solving logarithmic equations by working through examples and exercises. You can also try solving logarithmic equations on your own and then check your solutions to make sure they are correct.

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to Maria's equation and answer any questions you may have. We covered topics such as the properties of logarithmic equations, how to simplify the expression inside the logarithm, and how to check the solution to make sure it satisfies the equation. We hope this guide has been helpful in helping you understand Maria's equation and how to solve it.

Additional Resources

• [1] "Logarithmic Equations" by Khan Academy
• [2] "Solving Logarithmic Equations" by Mathway
• [3] "Logarithmic Equations" by Purplemath

FAQs

• Q: What is the solution to Maria's equation? A: The solution to Maria's equation is $x = \frac{5}{4}$.
• Q: What are the properties of logarithmic equations? A: The properties of logarithmic equations include the product rule and the quotient rule.
• Q: How do you simplify the expression inside the logarithm? A: To simplify the expression inside the logarithm, you can cancel out any common factors.
• Q: What are some common mistakes to avoid when solving logarithmic equations? A: Some common mistakes to avoid when solving logarithmic equations include not combining the logarithmic expressions using the product rule, not simplifying the expression inside the logarithm, not equating the logarithmic expression to the right-hand side of the equation, and not checking the solution to make sure it satisfies the equation.