1. **State the problem:** We need to find the sum of the first ten terms of the series $\log x + \log x^2 + \log x^3 + \cdots + \log x^{10}$. 2. **Use logarithm properties:** Recall that $\log a^b = b \log a$. So each term $\log x^k = k \log x$. 3. **Rewrite the series:** The series becomes $\log x + 2 \log x + 3 \log x + \cdots + 10 \log x$. 4. **Factor out $\log x$:** $$\log x (1 + 2 + 3 + \cdots + 10)$$ 5. **Sum of first 10 natural numbers:** $$1 + 2 + 3 + \cdots + 10 = \frac{10 \times (10+1)}{2} = \frac{10 \times 11}{2} = 55$$ 6. **Calculate the sum:** $$\log x \times 55 = 55 \log x$$ **Final answer:** $$\boxed{55 \log x}$$