1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \log \left(1 + x\right)^{\frac{1}{x}}$$. 2. **Rewrite the expression:** Using logarithm properties, $$\log \left(1 + x\right)^{\frac{1}{x}} = \frac{1}{x} \log(1 + x)$$. 3. **Recall the limit definition:** We know that $$\lim_{x \to 0} \frac{\log(1 + x)}{x} = 1$$. 4. **Apply the limit:** Therefore, $$\lim_{x \to 0} \frac{1}{x} \log(1 + x) = \lim_{x \to 0} \frac{\log(1 + x)}{x} = 1$$. 5. **Interpretation:** The original limit is the logarithm of the expression, so the limit of the expression itself is $$\lim_{x \to 0} \left(1 + x\right)^{\frac{1}{x}} = e^{1} = e$$. **Final answer:** $$e$$.