Answer
False
Work Step by Step
Check both sides:
$$\frac{\ln 4}{ \ln 8}=\frac{\ln 2^{2}}{ \ln 2^{3}}=\frac{2\ln 2}{3\ln 2}=\frac{2}{3}$$
$$\ln 4-\ln 8=\ln\frac{4}{8}=\ln\frac{1}{2}$$
Since $\frac{2}{3} \neq \ln\frac{1}{2}$ it follows that the statement is false.