Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)

log10

*xy*3

*z*6

To expand the expression \( \log_{10} (x y^3 z^6) \) using the properties of logarithms, let's proceed step-by-step.

### Step 1: Apply the Product Rule
The product rule states that \( \log_b (A \cdot B) = \log_b A + \log_b B \).

So,
\[
\log_{10} (x y^3 z^6) = \log_{10} x + \log_{10} y^3 + \log_{10} z^6
\]

### Step 2: Apply the Power Rule
The power rule states that \( \log_b (A^n) = n \cdot \log_b A \).

Applying this rule to each term with an exponent:
\[
\log_{10} x + 3 \cdot \log_{10} y + 6 \cdot \log_{10} z
\]

### Final Expanded Expression
Thus, the expanded form is:
\[
\log_{10} x + 3 \log_{10} y + 6 \log_{10} z
\]

Would you like further details or have any questions?

Here are some related questions to consider:
1. How would the expansion change if the expression were \( \log_{10} (x^2 y^3 z) \)?
2. What happens if one of the terms in the expression, such as \( y \), is raised to a different exponent?
3. Can the expression \( \log_{10} \frac{x y^3}{z^6} \) be expanded similarly?
4. What would be the approach if the base of the logarithm was different, say \( \log_2 \) instead of \( \log_{10} \)?
5. How does the product rule for logarithms relate to properties of exponents?

**Tip:** Remember that logarithmic properties allow for simplification, especially in complex expressions involving products, quotients, or powers of variables.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) log10(xy^3z^6)

Learn how to expand the logarithmic expression log10(xy^3z^6) using the product and power rules for logarithms. This step-by-step guide is perfect for high school students studying logarithms and algebra.

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