1. **State the problem:** Solve for $a$ in the equation $$55 = 25 + \left(\frac{5}{11}\right)^a.$$\n\n2. **Isolate the exponential term:** Subtract 25 from both sides to get $$55 - 25 = \left(\frac{5}{11}\right)^a.$$\n\n3. Simplify the left side: $$30 = \left(\frac{5}{11}\right)^a.$$\n\n4. **Rewrite the equation:** $$\left(\frac{5}{11}\right)^a = 30.$$\n\n5. **Take the natural logarithm of both sides:** $$\ln\left(\left(\frac{5}{11}\right)^a\right) = \ln(30).$$\n\n6. Use the logarithm power rule: $$a \ln\left(\frac{5}{11}\right) = \ln(30).$$\n\n7. **Solve for $a$:** $$a = \frac{\ln(30)}{\ln\left(\frac{5}{11}\right)}.$$\n\n8. **Evaluate the logarithms:**\n$$\ln(30) \approx 3.4012,$$\n$$\ln\left(\frac{5}{11}\right) = \ln(5) - \ln(11) \approx 1.6094 - 2.3979 = -0.7885.$$\n\n9. **Calculate $a$:** $$a \approx \frac{3.4012}{-0.7885} \approx -4.31.$$\n\n**Final answer:** $$a \approx -4.31.$$