https://file.aiquickdraw.com/mathbot/file/1727047728980wn1qaec5.jpg

The expression given in the image is:

\[
8 \log_5 \sqrt[4]{y}
\]

We can simplify this expression using logarithmic properties.

### Step 1: Simplify the radical
The expression inside the logarithm is \( \sqrt[4]{y} \), which can be written as \( y^{1/4} \). So the expression becomes:

\[
8 \log_5 y^{1/4}
\]

### Step 2: Apply the power rule of logarithms
The power rule of logarithms states that \( \log_b (x^n) = n \log_b x \). Applying this to the expression:

\[
8 \times \frac{1}{4} \log_5 y
\]

Simplifying this:

\[
2 \log_5 y
\]

Thus, the expression simplifies to:

\[
2 \log_5 y
\]

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Would you like further clarification or more details on logarithmic rules? Here are five related questions to explore this further:

1. What are the other key logarithmic properties used for simplifying expressions?
2. How do you convert logarithmic expressions with different bases?
3. Can you express \( \log_5 y^3 \) using logarithmic rules?
4. How do logarithms interact with exponents when solving equations?
5. How can you apply logarithmic differentiation to complex functions?

**Tip:** Always remember the power rule of logarithms: \( \log_b(x^n) = n \log_b(x) \), which simplifies logarithmic expressions with exponents!

Express the following as a sum, difference, or multiple of logarithms: 8 log_5 (4th root of y)

Explore the step-by-step simplification of the expression 8 log_5 (4th root of y) using logarithmic properties, particularly focusing on the power rule. This detailed guide is suitable for high school students learning about logarithms.

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