Which of the following is equivalent to ln(sqrt(x)/y³)?

This option correctly represents the simplified form of ln(sqrt(x)/y³). By applying the logarithmic properties, we find that ln(sqrt(x)) equals 1/2 ln x and ln(y³) equals 3 ln y. Combining these results confirms that the expression simplifies to -1/2 ln x - 3 ln y.

This choice incorrectly suggests that the logarithm of y³ contributes positively. The correct application of the properties of logarithms dictates a subtraction due to division, not addition. Thus, this option misrepresents the relationship between the terms.

This choice mistakenly applies the power rule to ln(y³), resulting in an incorrect coefficient for ln y. The correct simplification should maintain the coefficient of -3 for ln y, reinforcing that this option deviates from the correct properties of logarithmic functions.

This option also incorrectly applies the properties of logarithms, suggesting an addition of the terms rather than a subtraction. The division in the original expression requires a negative sign in front of ln y, making this option incorrect.