Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
log Subscript 3 Baseline left parenthesis StartFraction StartRoot y EndRoot Over 9 EndFraction right parenthesis
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Part 1
log Subscript 3 Baseline left parenthesis StartFraction StartRoot y EndRoot Over 9 EndFraction right parenthesis
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Answer 1

To expand the logarithmic expression \( \log_3\left(\frac{\sqrt{y}}{9}\right) \) using properties of logarithms, we can use the following properties:

1. **Quotient Rule:** \( \log_b\left(\frac{A}{B}\right) = \log_b(A) - \log_b(B) \)
2. **Product Rule:** \( \log_b(A \cdot B) = \log_b(A) + \log_b(B) \)
3. **Power Rule:** \( \log_b(A^c) = c \cdot \log_b(A) \)

Now, let's expand the given expression step-by-step.

### Step 1: Apply the Quotient Rule

\[
\log_3\left(\frac{\sqrt{y}}{9}\right) = \log_3(\sqrt{y}) - \log_3(9)
\]

### Step 2: Simplify \(\log_3(\sqrt{y})\)

Recall that \(\sqrt{y} = y^{1/2}\). Using the Power Rule:

\[
\log_3(\sqrt{y}) = \log_3(y^{1/2}) = \frac{1}{2} \log_3(y)
\]

### Step 3: Evaluate \(\log_3(9)\)

Recognize that \(9 = 3^2\). Therefore:

\[
\log_3(9) = \log_3(3^2) = 2 \log_3(3) = 2
\]

### Step 4: Combine Everything

Now substituting the values back in, we have:

\[
\log_3\left(\frac{\sqrt{y}}{9}\right) = \frac{1}{2} \log_3(y) - 2
\]

### Final Answer

So, the expanded form of \( \log_3\left(\frac{\sqrt{y}}{9}\right) \) is:

\[
\frac{1}{2} \log_3(y) - 2
\]