What's the integral of log(x) dx

In calculus, $\log(x)$ can sometimes imply the natural logarithm $\ln(x)$. However, since you just asked for $\ln(x)$, we will assume $\log(x)$ refers to the common logarithm, base 10, i.e., $\log_{10}(x)$.

We use the change of base formula: $\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$.

Since $\ln(10)$ is a constant, we can factor it out of the integral:

From the previous problem, we know that $\int \ln(x)\,dx = x\ln(x) - x + C$.

We can also express this using $\log_{10}(x)$ again:

The integral of $\log(x)\,dx$ (assuming base 10) is $\boxed{x\log_{10}(x) - \frac{x}{\ln(10)} + C}$.