Final answer:
The expression ln(a^2 b^3/cd) can be expanded, using logarithmic properties, to 2ln(a) + 3ln(b) - ln(c) - ln(d).
Step-by-step explanation:
To expand the natural logarithm expression ln(a^2 b^3/cd), we can apply the logarithm properties. The natural logarithm of a quotient is the difference of the logarithms, and the natural logarithm of a product is the sum of the logarithms. Furthermore, logarithms can be expanded when the argument is raised to a power by bringing the exponent out as a coefficient. Therefore, using these rules, we get:
ln(a^2 b^3/cd) = ln(a^2) + ln(b^3) - ln(c) - ln(d)
Applying the power rule of logarithms to ln(a^2) and ln(b^3), the expression can be further expanded to:
2ln(a) + 3ln(b) - ln(c) - ln(d)
This is the expanded form of the given logarithmic expression.