What Is The Value Of $\log_a\left(\frac{x Z^2}{y^{-2}}\right$\] When Given The Following Values?$\[ \begin{align*} \log_a(x) &= 2 \\ \log_a(y) &= 3 \\ \log_a(z) &= 4 \end{align*} \\]

$$

Introduction

In this article, we will explore the concept of logarithms and how to evaluate expressions involving logarithms. We will use the given values of $\log_a(x)$, $\log_a(y)$, and $\log_a(z)$ to find the value of $\log_a\left(\frac{x z^2}{y^{-2}}\right)$. This problem requires a deep understanding of logarithmic properties and how to apply them to evaluate complex expressions.

Understanding Logarithmic Properties

Before we dive into the problem, let's review some essential logarithmic properties:

• Product Rule: $\log_a(MN) = \log_a(M) + \log_a(N)$
• Quotient Rule: $\log_a\left(\frac{M}{N}\right) = \log_a(M) - \log_a(N)$
• Power Rule: $\log_a(M^p) = p \log_a(M)$

These properties will be crucial in evaluating the given expression.

Evaluating the Expression

Now, let's evaluate the expression $\log_a\left(\frac{x z^2}{y^{-2}}\right)$ using the given values of $\log_a(x)$, $\log_a(y)$, and $\log_a(z)$.

We can start by applying the Quotient Rule:

$$\log_a\left(\frac{x z^2}{y^{-2}}\right) = \log_a(x z^2) - \log_a(y^{-2})$$

Next, we can apply the Product Rule to the first term:

$$\log_a(x z^2) = \log_a(x) + \log_a(z^2)$$

Now, we can apply the Power Rule to the second term:

$$\log_a(z^2) = 2 \log_a(z)$$

Substituting the given values, we get:

$$\log_a(x z^2) = 2 + 2(4) = 10$$

Now, let's evaluate the second term:

$$\log_a(y^{-2}) = -2 \log_a(y)$$

Substituting the given value, we get:

$$\log_a(y^{-2}) = -2(3) = -6$$

Now, we can substitute these values back into the original expression:

$$\log_a\left(\frac{x z^2}{y^{-2}}\right) = 10 - (-6) = 16$$

Conclusion

In this article, we evaluated the expression $\log_a\left(\frac{x z^2}{y^{-2}}\right)$ using the given values of $\log_a(x)$, $\log_a(y)$, and $\log_a(z)$. We applied the Quotient Rule, Product Rule, and Power Rule to simplify the expression and find its value. The final answer is $\boxed{16}$.

Frequently Asked Questions

• What is the value of $\log_a(x)$, $\log_a(y)$, and $\log_a(z)$?
  • $\log_a(x) = 2$
  • $\log_a(y) = 3$
  • $\log_a(z) = 4$
• How do you evaluate the expression $\log_a\left(\frac{x z^2}{y^{-2}}\right)$?
  • Apply the Quotient Rule, Product Rule, and Power Rule to simplify the expression.
• What is the final answer?
  • $\boxed{16}$

Related Topics

• Logarithmic properties
• Evaluating expressions involving logarithms
• Logarithmic functions

References

• [1] "Logarithmic Properties" by Math Open Reference
• [2] "Evaluating Expressions Involving Logarithms" by Khan Academy
• [3] "Logarithmic Functions" by Wolfram MathWorld
  

Introduction

In our previous article, we explored the concept of logarithmic expressions and how to evaluate them using the given values of $\log_a(x)$, $\log_a(y)$, and $\log_a(z)$. In this article, we will address some of the most frequently asked questions related to logarithmic expressions.

Q&A

Q1: What is the value of $\log_a(x)$, $\log_a(y)$, and $\log_a(z)$?

A1: The values of $\log_a(x)$, $\log_a(y)$, and $\log_a(z)$ are given as:

• $\log_a(x) = 2$
• $\log_a(y) = 3$
• $\log_a(z) = 4$

Q2: How do you evaluate the expression $\log_a\left(\frac{x z^2}{y^{-2}}\right)$?

A2: To evaluate the expression $\log_a\left(\frac{x z^2}{y^{-2}}\right)$, you need to apply the following logarithmic properties:

• Quotient Rule: $\log_a\left(\frac{M}{N}\right) = \log_a(M) - \log_a(N)$
• Product Rule: $\log_a(MN) = \log_a(M) + \log_a(N)$
• Power Rule: $\log_a(M^p) = p \log_a(M)$

Q3: What is the final answer to the expression $\log_a\left(\frac{x z^2}{y^{-2}}\right)$?

A3: The final answer to the expression $\log_a\left(\frac{x z^2}{y^{-2}}\right)$ is $\boxed{16}$.

Q4: How do you apply the Quotient Rule to the expression $\log_a\left(\frac{x z^2}{y^{-2}}\right)$?

A4: To apply the Quotient Rule, you need to subtract the logarithm of the denominator from the logarithm of the numerator:

$$\log_a\left(\frac{x z^2}{y^{-2}}\right) = \log_a(x z^2) - \log_a(y^{-2})$$

Q5: How do you apply the Product Rule to the expression $\log_a(x z^2)$?

A5: To apply the Product Rule, you need to add the logarithms of the two factors:

$$\log_a(x z^2) = \log_a(x) + \log_a(z^2)$$

Q6: How do you apply the Power Rule to the expression $\log_a(z^2)$?

A6: To apply the Power Rule, you need to multiply the logarithm of the base by the exponent:

$$\log_a(z^2) = 2 \log_a(z)$$

Q7: What is the value of $\log_a(y^{-2})$?

A7: The value of $\log_a(y^{-2})$ is $-2 \log_a(y)$.

Q8: How do you substitute the values of $\log_a(x)$, $\log_a(y)$, and $\log_a(z)$ into the expression?

A8: To substitute the values, you need to replace the variables with their corresponding values:

$$\log_a(x z^2) = 2 + 2(4) = 10$$

$$\log_a(y^{-2}) = -2(3) = -6$$

Q9: What is the final answer after substituting the values?

A9: The final answer after substituting the values is $\boxed{16}$.

Conclusion

In this article, we addressed some of the most frequently asked questions related to logarithmic expressions. We provided step-by-step explanations and examples to help you understand the concepts and apply them to evaluate logarithmic expressions.

Related Topics

• Logarithmic properties
• Evaluating expressions involving logarithms
• Logarithmic functions

References

• [1] "Logarithmic Properties" by Math Open Reference
• [2] "Evaluating Expressions Involving Logarithms" by Khan Academy
• [3] "Logarithmic Functions" by Wolfram MathWorld