1. **Problem Statement:** Calculate the integral of the logarithm function $\int \log(x) \, dx$. 2. **Formula and Rules:** We use integration by parts, where $\int u \, dv = uv - \int v \, du$. 3. **Choose parts:** Let $u = \log(x)$ and $dv = dx$. 4. **Derivatives and integrals:** Then $du = \frac{1}{x} dx$ and $v = x$. 5. **Apply integration by parts:** $$\int \log(x) \, dx = x \log(x) - \int x \cdot \frac{1}{x} \, dx = x \log(x) - \int 1 \, dx$$ 6. **Simplify integral:** $$x \log(x) - x + C$$ 7. **Final answer:** $$\int \log(x) \, dx = x \log(x) - x + C$$ This means the integral of $\log(x)$ with respect to $x$ is $x \log(x) - x$ plus a constant of integration.