In the answer to this question Eric Näslund showed that logarithms can be written as the following limit of a sum:
$$\displaystyle \log(x) = \lim_{k\to \infty } \, \sum\limits_{n=k}^{x k} \frac{1}{n}$$
Changing the symbol for the sum $\sum$ into an integral $\int$ with the same limits as in the sum, one still gets logarithms:
$$\displaystyle \log(x) = \lim_{k\to \infty } \, \int\limits_k^{x k} \frac{1}{n} \, dn$$
or at least in Mathematica, so I guess it is true.
Are there other sums and integrals such that the function to be integrated or summed is the same, as well as the integration and summation limits, that give the same result? This provided that integrals or sums giving zero as result are not considered.
As a Mathematica program this is:
Clear[n, k, x];
Table[Limit[Integrate[1/n, {n, k, x*k}], k -> Infinity], {x, 1, 12}]
Table[Limit[Sum[1/n, {n, k, x*k}], k -> Infinity], {x, 1, 12}]
which gives the output:
{0, Log[2], Log[3], Log[4], Log[5], Log[6], Log[7], Log[8], Log[9], Log[10], Log[11], Log[12]}
in both cases.
Edit 7.4.2013: I now realized that the integral is not dependent on that the limit goes to infinity. Actually any value of k will give the same logarithm.
A small Mathematica program to illustrate this:
Table[Integrate[1/n, {n, k, 2*k}], {k, 1, 12}]
which gives:
{Log[2], Log[2], Log[2], Log[2], Log[2], Log[2], Log[2],
Log[2], Log[2], Log[2], Log[2], Log[2]}
