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The given expression is:

\[
\log_{y_2}{x_2^3} \cdot \log_{z_2}{y_2^3} \cdot \log_{x_2}{z_2^3}
\]

We need to simplify this product of logarithms. The logarithmic identity we can apply here is:

\[
\log_b{a^c} = c \cdot \log_b{a}
\]

Using this identity for each term:

1. \(\log_{y_2}{x_2^3} = 3 \cdot \log_{y_2}{x_2}\)
2. \(\log_{z_2}{y_2^3} = 3 \cdot \log_{z_2}{y_2}\)
3. \(\log_{x_2}{z_2^3} = 3 \cdot \log_{x_2}{z_2}\)

Now the expression becomes:

\[
3 \cdot \log_{y_2}{x_2} \cdot 3 \cdot \log_{z_2}{y_2} \cdot 3 \cdot \log_{x_2}{z_2}
\]

This simplifies further to:

\[
27 \cdot \log_{y_2}{x_2} \cdot \log_{z_2}{y_2} \cdot \log_{x_2}{z_2}
\]

Next, we can recognize that the terms \(\log_{y_2}{x_2} \cdot \log_{z_2}{y_2} \cdot \log_{x_2}{z_2}\) form a cycle of logarithms in different bases. By the **cyclic logarithm property** (which states that the product of logs with cyclic bases is equal to 1), we know:

\[
\log_{y_2}{x_2} \cdot \log_{z_2}{y_2} \cdot \log_{x_2}{z_2} = 1
\]

Thus, the final simplified result is:

\[
27
\]

Let me know if you need further clarification or a more detailed explanation!

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Here are 5 related questions:

1. How do you simplify logarithmic expressions involving powers?
2. What is the change of base formula for logarithms?
3. Can you explain the cyclic property of logarithms in more detail?
4. How does the product of logarithms behave with different bases?
5. How can logarithmic identities help in solving complex equations?

**Tip:** Always look for logarithmic identities like the power rule or cyclic properties when simplifying complex logarithmic expressions.

If log_y2(x2^3) * log_z2(y2^3) * log_x2(z2^3) = ?

Learn how to simplify the expression log_y2(x2^3) * log_z2(y2^3) * log_x2(z2^3) using logarithmic identities like the power rule and the cyclic property of logarithms. This step-by-step solution is suitable for advanced students in Grades 11-12.

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