xpand the logarithm fully using the properties of logs. Express the final answer in terms of log, xlog*x*, and log, ylog*y*.

log, start fraction, x, cubed, divided by, y, end fraction

log*yx*3​

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!

Expand the logarithm fully using the properties of logs. Express the final answer in terms of log(x) and log(y): log(x^3 / y).

This page explains how to fully expand the logarithmic expression log(x^3 / y) using the quotient and power rules. Learn step-by-step how to express the answer in terms of log(x) and log(y), ideal for students in Grades 10-12 studying logarithms and exponents.

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