Use the properties of logarithms to CONDENSE the expression. $$ 3 \ln 2+4 \ln x-\ln 5-2 \ln y= $$

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To condense the expression $$ 3 \ln 2 + 4 \ln x - \ln 5 - 2 \ln y $$, we will use the properties of logarithms:

1. **Power Rule**: $$ a \ln b = \ln(b^a) $$
2. **Product Rule**: $$ \ln a + \ln b = \ln(ab) $$
3. **Quotient Rule**: $$ \ln a - \ln b = \ln\left(\frac{a}{b}\right) $$

Now, let's apply these rules step by step:

1. **Apply the Power Rule**:
   - $$ 3 \ln 2 = \ln(2^3) = \ln(8) $$
   - $$ 4 \ln x = \ln(x^4) $$
   - $$ -2 \ln y = \ln(y^{-2}) $$

So, we can rewrite the expression as:
   $$
   \ln(8) + \ln(x^4) - \ln(5) + \ln(y^{-2})
   $$

2. **Combine using the Product Rule**:
   - Combine $$ \ln(8) + \ln(x^4) $$ to get $$ \ln(8x^4) $$.
   - Now we have $$ \ln(8x^4) - \ln(5) + \ln(y^{-2}) $$.

3. **Apply the Quotient Rule**:
   - Combine $$ \ln(8x^4) - \ln(5) $$ to get $$ \ln\left(\frac{8x^4}{5}\right) $$.

4. **Combine with $$ \ln(y^{-2}) $$**:
   - Now we have $$ \ln\left(\frac{8x^4}{5}\right) + \ln(y^{-2}) $$.
   - This can be combined using the Product Rule again:
   $$
   \ln\left(\frac{8x^4}{5} \cdot y^{-2}\right) = \ln\left(\frac{8x^4}{5y^2}\right)
   $$

Thus, the condensed expression is:
$$
\ln\left(\frac{8x^4}{5y^2}\right)
$$
The condensed expression is $$ \ln\left(\frac{8x^4}{5y^2}\right) $$.

Use the properties of logarithms to CONDENSE the expression. \( 3 \ln 2+4 \ln x-\ln 5-2 \ln y= \)

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