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.