Question 28

ln(sqrt(x)/y³) is equivalent to 1/2 ln x - 3 ln y.

The expression ln(sqrt(x)/y³) can be simplified using logarithmic properties, leading to the equivalent form of 1/2 ln x - 3 ln y.

A) -1/2 ln x - 3 ln y

This option incorrectly represents the coefficients of ln x. The negative sign before the 1/2 ln x is not consistent with the properties of logarithms when simplifying the original expression.

B) -1/2 ln x + 3 ln y

This option also incorrectly modifies the coefficient of ln y, changing its sign, which does not align with the logarithmic properties involved in the simplification of ln(sqrt(x)/y³).

C) 1/2 ln x - 3 ln y

This option correctly applies the properties of logarithms: ln(sqrt(x)) simplifies to (1/2)ln(x) and ln(y³) simplifies to 3ln(y). Thus, the overall expression simplifies to 1/2 ln x - 3 ln y, making this option correct.

D) 1/2 ln x - 3/2 ln y

While this option correctly represents the term for ln x, it incorrectly states the coefficient for ln y. The logarithmic property would yield a coefficient of -3, not -3/2, which makes this option incorrect.

E) 1/2 ln x + 3 ln y

This option is entirely incorrect as it misrepresents both the sign and coefficient of ln y. The simplification of the original expression should yield a negative coefficient for ln y, making this option invalid.

Conclusion

The option C, 1/2 ln x - 3 ln y, is definitively correct as it accurately reflects the simplifications derived from the properties of logarithms applied to the original expression. All other options fail to maintain the correct coefficients or signs, thereby misrepresenting the derived expression.