Final answer:
The question has a mistake because ln(5y−9y) simplifies to a logarithm of a negative number, which is undefined. The correct expansion using the quotient rule for a properly defined expression like ln(5y/9y) would be ln(5) − ln(9), after the y's cancel out. The quotient rule states that ln(a/b) = ln(a) − ln(b).
Step-by-step explanation:
The student seems to want to use the quotient rule for logarithms to expand ln(5y−9y). However, there is a mistake in the expression because it simplifies to −4y, and you cannot take the natural logarithm of a negative number if y is positive. Assuming there's a typo, and the correct expression has different terms in the parenthesis, such as ln(5y/9y), you can use the quotient rule to expand this logarithm.
The quotient rule states that the logarithm of a quotient is the difference of the logarithms: ln(a/b) = ln(a) − ln(b). Applying this to ln(5y/9y), we get ln(5) − ln(9) because the y's cancel out.
It's important to remember that the logarithm of a negative number is undefined in the realm of real numbers, and that the quotient rule only applies when all components are positive and defined.
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