1. **State the problem:** Simplify the expression $$\frac{\ln(2t) - \ln(8)}{\ln(3) - \ln(2t) + \ln(4)}$$. 2. **Recall logarithm rules:** - $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ - $\ln(a) + \ln(b) = \ln(ab)$ 3. **Simplify numerator:** $$\ln(2t) - \ln(8) = \ln\left(\frac{2t}{8}\right) = \ln\left(\frac{t}{4}\right)$$ 4. **Simplify denominator:** $$\ln(3) - \ln(2t) + \ln(4) = \ln(3) + \ln(4) - \ln(2t) = \ln(12) - \ln(2t) = \ln\left(\frac{12}{2t}\right) = \ln\left(\frac{6}{t}\right)$$ 5. **Rewrite the entire expression:** $$\frac{\ln\left(\frac{t}{4}\right)}{\ln\left(\frac{6}{t}\right)}$$ 6. **Final answer:** $$\boxed{\frac{\ln\left(\frac{t}{4}\right)}{\ln\left(\frac{6}{t}\right)}}$$ This is the simplified form of the given expression.