Infinite sum of logarithms Is there any closed form for this expression
$$
\sum_{n=0}^\infty\ln(n+x)
$$

As others correctly mentioned, the expression diverges. Yet, if necessary, you can get quite good asymptotics:
$$
\sum_{k=1}^{n} \log (k+x) = \sum_{k=1}^{n} \log k + \sum_{k=1}^{n} \log (1+ \frac{x}{k}) \sim n \log n + \sum_{k=1}^{n} \frac{x}{k} = n \log n + x \log n \\
=(n+x) \log n 
$$ 
