ANSWER
[tex]c. \ln( \frac{2 {x}^{3} }{3y}) [/tex]
EXPLANATION
We want to write
[tex] ln(2x) + 2 ln(x) - ln(3y) [/tex]
as a single logarithm.
Use the power rule to rewrite the middle term:
[tex]n \: ln(a) = ln( {a}^{n} ) [/tex]
[tex]ln(2x) + ln( {x}^{2} ) - ln(3y) [/tex]
Use the product rule to obtain:
[tex]ln(a) + ln(b) = ln(ab) [/tex]
[tex]ln(2x \times {x}^{2} )- ln(3y) [/tex]
[tex]ln(2{x}^{3} )- ln(3y)[/tex]
Apply the quotient rule:
[tex] ln(a) - ln(b) = ln( \frac{a}{b})[/tex]
[tex]ln(2{x}^{3} )- ln(3y) =\ln(\frac{2 {x}^{3} }{3y})[/tex]
