# What function has derivative $\log(x)$?

I have the derivative of a function $f(x)$: $f'(x) = \log(x)$, where $\log(x)$ is the natural logarithm. What's the original function $f(x)$ and what is that calculation called in English?

• This is called "integrating $\log(x)$". Mar 26, 2013 at 14:19

Hint : integrate by parts. $du = dx$, $v=\log{x}$.

$$\int \log{x} \, dx = x \log{x} - \int x \frac{d}{dx} (\log{x}) = x \log{x} - x + C$$

• Ah, integration it's called. But I'm a newbie, so I cannot follow the rest of your answer. Can you explain?
– user63495
Mar 26, 2013 at 14:18
• Integration by parts, there are some examples
– Blex
Mar 26, 2013 at 14:52

$[x \ln x -x\ ] + C$ is what you're looking for, it's called the anti-derivative of $\log x$

• ...plus a constant....:) Mar 26, 2013 at 14:43
• obviously... =)
– Bob
Mar 26, 2013 at 15:33

$x \log x - x$ try differentiating this.

Here's a way to think about it if you haven't been introduced to integration.

\begin{aligned} \displaystyle \log{x} = (\log{x}+1)-1 = (x)'\log{x}+x(\log{x})'-1+0 = (x\log{x})'-(x)'+(\mathcal{C})' = (x\log{x}-x+\mathcal{C})' \end{aligned}

Where $\mathcal{C}$ is any constant. Therefore your function is $f(x) = x\log{x}-x+\mathcal{C}$.