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The given expression is:

\[
\frac{3}{2} \ln(4x^8) - \frac{1}{5} \ln(2y^{30})
\]

### Step 1: Apply logarithmic properties
First, use the power rule of logarithms: \( a \ln(b) = \ln(b^a) \). This allows us to move the coefficients (3/2 and 1/5) as exponents inside the logarithms.

\[
\ln((4x^8)^{\frac{3}{2}}) - \ln((2y^{30})^{\frac{1}{5}})
\]

### Step 2: Simplify exponents
Now, simplify the exponents inside the logarithms.

- \( (4x^8)^{\frac{3}{2}} = 4^{\frac{3}{2}} \cdot (x^8)^{\frac{3}{2}} = 8x^{12} \) because \( 4^{\frac{3}{2}} = 8 \) and \( (x^8)^{\frac{3}{2}} = x^{12} \).
- \( (2y^{30})^{\frac{1}{5}} = 2^{\frac{1}{5}} \cdot (y^{30})^{\frac{1}{5}} = 2^{\frac{1}{5}} \cdot y^6 \).

The expression becomes:

\[
\ln(8x^{12}) - \ln(2^{\frac{1}{5}} y^6)
\]

### Step 3: Use the logarithmic subtraction rule
Next, apply the rule \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \).

\[
\ln\left(\frac{8x^{12}}{2^{\frac{1}{5}} y^6}\right)
\]

### Final Answer:
Thus, the expression simplifies to:

\[
\boxed{\ln\left(\frac{8x^{12}}{2^{\frac{1}{5}} y^6}\right)}
\]

Would you like further details on any of the steps, or do you have any questions?

Here are 5 related questions:
1. How do you apply logarithmic rules for multiplication and division?
2. What is the general power rule for logarithms?
3. How would you simplify a natural logarithm involving roots?
4. Can this expression be simplified further with specific values for \(x\) and \(y\)?
5. How do logarithmic properties help in solving exponential equations?

**Tip:** Always check if terms inside logarithms can be combined or simplified before applying logarithmic rules.

Express as a single logarithm and, if possible, simplify: (3/2)ln(4x^8) - (1/5)ln(2y^30)

Learn how to express the logarithmic expression (3/2)ln(4x^8) - (1/5)ln(2y^30) as a single logarithm using properties of logarithms. This problem covers key mathematical concepts such as the power rule and subtraction rule of logarithms. Suitable for students in Grades 10-12.

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