https://file.aiquickdraw.com/mathbot/file/1726119625016qfmct6vl.jpg

The given problem is asking to express the logarithm of the expression \(\log \left( \sqrt{\frac{x^3}{\sqrt{y}}} \right)\) in terms of \(p = \log x\) and \(q = \log y\).

### Step-by-step solution:

1. **Start with the given expression:**
   \[
   \log \left( \sqrt{\frac{x^3}{\sqrt{y}}} \right)
   \]

2. **Simplify the square root expression:**
   We know that \(\sqrt{A} = A^{1/2}\), so we can rewrite the expression:
   \[
   \sqrt{\frac{x^3}{\sqrt{y}}} = \left( \frac{x^3}{y^{1/2}} \right)^{1/2}
   \]

3. **Use the logarithmic property \(\log(a^b) = b \log(a)\):**
   \[
   \log \left( \left( \frac{x^3}{y^{1/2}} \right)^{1/2} \right) = \frac{1}{2} \log \left( \frac{x^3}{y^{1/2}} \right)
   \]

4. **Apply the property \(\log \left( \frac{a}{b} \right) = \log a - \log b\):**
   \[
   \frac{1}{2} \left( \log(x^3) - \log(y^{1/2}) \right)
   \]

5. **Simplify using \(\log(x^n) = n \log x\):**
   \[
   \frac{1}{2} \left( 3 \log x - \frac{1}{2} \log y \right)
   \]

6. **Substitute \(p = \log x\) and \(q = \log y\):**
   \[
   \frac{1}{2} \left( 3p - \frac{1}{2}q \right)
   \]

7. **Simplify the expression:**
   \[
   \frac{1}{2} \left( 3p - \frac{q}{2} \right) = \frac{3p}{2} - \frac{q}{4}
   \]

### Final answer:
\[
\log \left( \sqrt{\frac{x^3}{\sqrt{y}}} \right) = \frac{3p}{2} - \frac{q}{4}
\]

Would you like further clarification or details on any step?

---

Here are 5 related questions to explore:

1. How do logarithmic properties help simplify expressions?
2. Can you express \(\log \left( x^4y^2 \right)\) in terms of \(p\) and \(q\)?
3. What is the logarithmic expression for \(\log \left( \frac{x^2}{y^3} \right)\) in terms of \(p\) and \(q\)?
4. How do you simplify nested radicals using logarithms?
5. What other logarithmic identities are useful for solving complex expressions?

**Tip:** Always keep logarithmic properties in mind, such as \(\log(a^b) = b \log(a)\), as they help break down complex terms into simpler components.

Diketahui, log x = p, log y = q. Nyatakan log(√(x^3 / √y)) dalam bentuk p dan q.

Learn how to express log(√(x^3 / √y)) in terms of p = log(x) and q = log(y) using logarithmic properties. This problem involves simplifying complex expressions and applying key logarithmic rules. Ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation