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We are given the expression:

\[
\frac{1}{3} (\log x - 2 \log y + 3 \log z)
\]

Let's go step-by-step to simplify it:

### Step 1: Distribute the \(\frac{1}{3}\)

We will distribute \(\frac{1}{3}\) to each term inside the parentheses:

\[
\frac{1}{3} \cdot \log x - \frac{1}{3} \cdot 2 \log y + \frac{1}{3} \cdot 3 \log z
\]

### Step 2: Simplify each term

Simplify the multiplication:

\[
\frac{1}{3} \log x - \frac{2}{3} \log y + \log z
\]

### Step 3: Combine the logarithmic terms

Using logarithmic properties:

- \(\log a - \log b = \log \left(\frac{a}{b}\right)\)
- \(n \log a = \log a^n\)

We can rewrite the expression as:

\[
\log x^{\frac{1}{3}} - \log y^{\frac{2}{3}} + \log z
\]

Now apply the logarithmic subtraction and addition properties:

\[
\log \left(\frac{x^{\frac{1}{3}} z}{y^{\frac{2}{3}}}\right)
\]

Thus, the simplified expression is:

\[
\log \left(\frac{x^{\frac{1}{3}} z}{y^{\frac{2}{3}}}\right)
\]

This is the final simplified form of the given expression.

Would you like further details on any step?

### Follow-up Questions:
1. What are the properties of logarithms used to simplify this expression?
2. Can you simplify an expression like \(\log (a^2 b^3 / c)\)?
3. How do the logarithmic laws work with fractional exponents?
4. What is the relationship between logarithmic addition and multiplication?
5. Can we express this logarithmic expression using natural logarithms?

### Tip:
Remember, logarithmic properties are powerful tools for simplifying complex expressions. Always check for opportunities to apply them!

Can you show the steps to simplify the expression 1/3 (log x - 2 log y + 3 log z)?

Learn how to simplify the logarithmic expression 1/3 (log x - 2 log y + 3 log z) using key properties of logarithms. This detailed step-by-step guide explains how to distribute the coefficient, apply logarithmic rules, and combine the terms. Suitable for students in Grades 10-12.

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