Final answer:
The expression 5 ln x + 4 ln y as a single logarithm should be ln(x^5y^4), using logarithmic properties of exponents and multiplication. However, this is not listed, the closest option, assuming a typo, would be C) ln(5xy^4), or D) if mislabeled because options A and B do not apply the logarithm rules correctly.
Step-by-step explanation:
To express the expression 5 ln x + 4 ln y as a single logarithm, we should apply a few logarithmic properties regarding exponents and multiplication. First, the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number allows us to rewrite the expression as ln(x5) + ln(y4). Then, we make use of the property that the logarithm of the product of two numbers is the sum of the logarithms of the two numbers, combining the two terms into a single logarithm: ln(x5y4).
Therefore, the correct expression given the options provided is ln(x5y4), which is not explicitly listed in the choices, but the closest correct option, considering typical simplifications in math problems, would be C) ln(5xy4) if interpreted as a typo, or option D) if mislabeled. Options A and B ignore the logarithmic properties of exponents and multiplication.