Answer:
[tex]\dfrac{1}{3}\ln x -\dfrac{1}{3}\ln y[/tex]
Step-by-step explanation:
Given logarithmic expression:
[tex]\ln\sqrt[3]{\dfrac{x}{y}}[/tex]
To expand the given logarithmic expression, we can use the properties of logarithms:
[tex]\boxed{\begin{array}{c}\underline{\textsf{Properties of Logarithms}}\\\\\textsf{Power:}\;\;\ln x^n=n\ln x\\\\\textsf{Quotient:}\;\;\ln \left(\dfrac{x}{y}\right)=\ln x - \ln y\end{array}}[/tex]
Rewrite the cube root as a fractional exponent:
[tex]\ln\left(\dfrac{x}{y}\right)^{\frac{1}{3}}[/tex]
Now, apply the power law of logarithms, which states that the logarithm of a power of a number is equal to the exponent multiplied by the logarithm of the number:
[tex]\dfrac{1}{3}\ln\left(\dfrac{x}{y}\right)[/tex]
Finally, apply the quotient law of logarithms, which states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator:
[tex]\dfrac{1}{3}\left(\ln x -\ln y\right)[/tex]
[tex]\dfrac{1}{3}\ln x -\dfrac{1}{3}\ln y[/tex]
Therefore, the expansion of the given logarithmic expression is:
[tex]\Large\boxed{\boxed{\dfrac{1}{3}\ln x -\dfrac{1}{3}\ln y}}[/tex]
