Condense the expression to the logarithm of a single quantity.
log x − 2 log y + 3 log z

To condense the expression \( \log x - 2 \log y + 3 \log z \) into the logarithm of a single quantity, we can use the properties of logarithms:

1. **Power Rule**: \( a \log b = \log(b^a) \).
2. **Product Rule**: \( \log a + \log b = \log(ab) \).
3. **Quotient Rule**: \( \log a - \log b = \log\left(\frac{a}{b}\right) \).

### Steps:
1. Apply the power rule to \( -2 \log y \) and \( 3 \log z \):
   \[
   \log x - 2 \log y + 3 \log z = \log x - \log(y^2) + \log(z^3)
   \]

2. Now, use the product and quotient rules:
   \[
   \log x - \log(y^2) + \log(z^3) = \log \left(\frac{x \cdot z^3}{y^2}\right)
   \]

### Final Answer:
\[
\log \left(\frac{x \cdot z^3}{y^2}\right)
\]

Would you like a more detailed breakdown or have any questions?

---

Here are five related questions to practice:

1. How would you condense \( 4 \log a + \log b - \frac{1}{2} \log c \)?
2. Simplify \( \log(x^2) + 3 \log(y) - \log(z) \).
3. Write \( \log(a) - \log(b) + 2 \log(c) \) as a single logarithm.
4. If given \( 2 \log x - 5 \log y + \log z \), how would you condense it?
5. How can you expand \( \log\left(\frac{a^2 b}{c^3}\right) \)?

**Tip:** Remember that applying the power rule to convert coefficients is often the first step in simplifying complex logarithmic expressions.

Condense the expression to the logarithm of a single quantity: log x − 2 log y + 3 log z

Discover how to condense the expression log x − 2 log y + 3 log z into a single logarithm. This problem illustrates key logarithmic properties, including the power, product, and quotient rules, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation