To fully expand the logarithmic expression $\log\left(\frac{x^3}{y}\right)$, we can use the properties of logarithms.

The two main properties we'll use are:

1. Quotient Rule: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$
2. Power Rule: $\log(a^n) = n\log(a)$

Let's expand $\log\left(\frac{x^3}{y}\right)$ step by step:

1. Apply the Quotient Rule:
   $\log\left(\frac{x^3}{y}\right) = \log(x^3) - \log(y)$

2. Apply the Power Rule to $\log(x^3)$:
   $\log(x^3) = 3\log(x)$

3. Substitute this back into the expression: $3\log(x) - \log(y)$

So, the fully expanded expression is: $3\log(x) - \log(y)$

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Would you like more details on logarithm properties, or do you have any other questions?

Here are 5 related questions to expand your understanding:

1. How would you simplify $\log\left(\frac{x^4}{z^2}\right)$?
2. What is the value of $\log(10^5)$ using the power rule?
3. Can you expand $\log\left(\frac{xy^2}{z^3}\right)$?
4. How does the change of base formula work with logarithms?
5. What is the logarithm of a product, and how is it expanded?

Tip: Always remember to apply the power rule before the quotient rule when simplifying logarithmic expressions involving powers!