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!