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To simplify the expression \( \log_5(\sqrt{y}) \) using the power property, we can proceed as follows:

### Solution
1. Start with the expression:
   \[
   \log_5(\sqrt{y})
   \]
   
2. Recognize that \( \sqrt{y} \) can be rewritten as \( y^{1/2} \). Thus:
   \[
   \log_5(\sqrt{y}) = \log_5(y^{1/2})
   \]
   
3. Apply the **power property** of logarithms, which states that \( \log_b(x^c) = c \cdot \log_b(x) \). Here, \( c = \frac{1}{2} \):
   \[
   \log_5(y^{1/2}) = \frac{1}{2} \cdot \log_5(y)
   \]

4. The simplified form is:
   \[
   \frac{1}{2} \cdot \log_5(y)
   \]

### Answer Format
According to the instructions, the answer format should be in lowercase with spaces being optional. So the answer can be written as:

\[
\frac{1}{2} \log_5(y)
\]

Or in the format specified by the instructions:

**Answer:** `1/2 log_5(y)`

Would you like more details on this solution or have any questions?

### Additional Practice Questions

1. Simplify \( \log_2(\sqrt{x}) \) using the power property.
2. Rewrite \( \log_3(x^{1/3}) \) in its simplified form.
3. Express \( \log_7(\frac{1}{y}) \) using logarithmic properties.
4. Simplify \( \log_{10}(\sqrt{a \cdot b}) \) using the power and product properties.
5. Rewrite \( \log_5((x^2)^{1/4}) \) in its simplest form.

### Tip
When simplifying logarithmic expressions with roots, remember to convert roots to fractional exponents (e.g., \( \sqrt{y} = y^{1/2} \)), which makes it easier to apply logarithmic properties.

Use the power property to rewrite the expression log_5(√y). Assume all variables are positive.

Learn how to simplify the expression log_5(√y) by applying the power property of logarithms. This step-by-step solution explains how to rewrite the square root in exponential form and then apply the power rule for logarithms. Ideal for students in Grades 9-11.

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