Answer
$\dfrac{1}{4} (a-b)$
Work Step by Step
$\ln \left(\sqrt[4]{\dfrac{2}{3}} \right) = \ln \left(\dfrac{2}{3} \right)^{\dfrac{1}{4}}$
With $\log_a M^r = r \log_a M$, then
$\ln \left(\dfrac{2}{3} \right)^{\dfrac{1}{4}} = \dfrac{1}{4} \ln \left(\dfrac{2}{3} \right)$
Recall that $\log_a\left(\dfrac{M}{N}\right) = \log_a M-\log_a N$.
Using the rule above gives:
$\ln \left(\dfrac{2}{3} \right) = \ln2-\ln3$
With $\because \ln3 = b \hspace{20pt} \text{and} \hspace{20pt} \ln2=a$, then
$\ln2-\ln3 = a-b$
Thus,
$\ln \left(\dfrac{2}{3} \right) = a-b$
Therefore,
$\ln \left(\sqrt[4]{\dfrac{2}{3}} \right) = \dfrac{1}{4}\ln{\left(\dfrac{2}{3}\right)} = \boxed{\dfrac{1}{4} (a-b)}$
