1. **Stating the problem:** Solve the equation $$\log_{\sqrt{27}} m = 2 \frac{2}{3}$$ for $m$. 2. **Recall the logarithm definition:** For $\log_a b = c$, it means $$a^c = b$$. 3. **Rewrite the base:** Note that $$\sqrt{27} = 27^{1/2} = (3^3)^{1/2} = 3^{3/2}$$. 4. **Rewrite the equation:** $$\log_{3^{3/2}} m = \frac{8}{3}$$ (since $2 \frac{2}{3} = \frac{8}{3}$). 5. **Apply the definition:** $$\left(3^{3/2}\right)^{8/3} = m$$. 6. **Simplify the exponent:** $$3^{(3/2) \times (8/3)} = 3^{4} = 81$$. 7. **Final answer:** $$m = 81$$.