What is the effect of the natural logarithm in the frequency domain?
First note that when you use a logarithmic function you shall avoid negative arguments if it's output should be real valued. Then consider the following relation: 
$$ y[n] = \ln( 1 + x[n] )$$
where the input $x[n]$ is conditioned such that $ 0 &lt; 1 + x[n] \leq2$ is maintained. Then a Maclaurin series expansion of this expression will be

$$ y[n] = \ln(1 + x[n]) = \sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k} x^k[n] .$$

For very small $x[n]$ the following approximation can also be taken: 
$$ y[n] \approx x[n] - x^2[n]/2 $$

Applying DTFT on this last line yields:

$$ Y(e^{j\omega}) \approx X(e^{j\omega}) - \frac{1/2}{2\pi} \left( X(e^{j\omega}) \star X(e^{j\omega}) \right) $$ 

The spectrum, $Y(e^{j\omega})$, of the logarithm of $1+x[n]$ will include multiple convolutions of $X(e^{j\omega})$ with itself. The relation becomes exact (with added terms which were ignored) as $k$ approaches infinity, showing that the spectrum will be broadened. 

Indeed for any nonzero bandwidth input $x[n]$, the exact output $y[n]$ would have infinite bandwidth (or equivalently unavoidable aliasing) and no finite upsampling is enough to properly represent it. However in practice as $|x[n]|&lt;1$ is maintained, its high powers, $x^k$ will tend to go zero quickly and can be ignored compared to some suitable threshold...