Question

Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
log Subscript b Baseline left parenthesis StartFraction x cubed y Over z Superscript 6 EndFraction right parenthesis
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Part 1
log Subscript b Baseline left parenthesis StartFraction x cubed y Over z Superscript 6 EndFraction right parenthesis
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Answer 1

To expand the logarithmic expression \( \log_b \left( \frac{x^3 y}{z^6} \right) \), we will use the properties of logarithms, specifically:

1. **Logarithm of a quotient**: \( \log_b \left( \frac{A}{B} \right) = \log_b(A) - \log_b(B) \)
2. **Logarithm of a product**: \( \log_b(AB) = \log_b(A) + \log_b(B) \)
3. **Logarithm of a power**: \( \log_b(A^n) = n \cdot \log_b(A) \)

Now, applying these properties step-by-step:

1. **Use the quotient rule**:
\[
\log_b \left( \frac{x^3 y}{z^6} \right) = \log_b(x^3 y) - \log_b(z^6)
\]

2. **Use the product rule on \( \log_b(x^3 y) \)**:
\[
\log_b(x^3 y) = \log_b(x^3) + \log_b(y)
\]

3. **Use the power rule on \( \log_b(x^3) \)**:
\[
\log_b(x^3) = 3 \cdot \log_b(x)
\]

4. **Use the power rule on \( \log_b(z^6) \)**:
\[
\log_b(z^6) = 6 \cdot \log_b(z)
\]

Now we can put it all together:
\[
\log_b \left( \frac{x^3 y}{z^6} \right) = \log_b(x^3 y) - \log_b(z^6) = \left(3 \cdot \log_b(x) + \log_b(y)\right) - (6 \cdot \log_b(z))
\]

So, the expanded expression is:
\[
3 \cdot \log_b(x) + \log_b(y) - 6 \cdot \log_b(z)
\]

Thus, the final answer is:
\[
\log_b \left( \frac{x^3 y}{z^6} \right) = 3 \cdot \log_b(x) + \log_b(y) - 6 \cdot \log_b(z)
\]