Since $\ln x = - \ln(1/x)$, to evaluate the logarithm near zero is equivalent to evaluating it for large argument. You can then use the result $$\ln y=\frac{\pi}{2a}\left(1+{\cal O}(y^{-2})\right),$$
with $a$ the <a href="https://en.wikipedia.org/wiki/Arithmetic–geometric_mean">arithmetic–geometric mean</a>$^\ast$ of $a_0=1$ and $b_0=4/y$, see page 11 of <A HREF="https://arxiv.org/abs/1004.3412">Multiple-precision zero-finding methods and the complexity of elementary function evaluation.</A>

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$^\ast$ Starting from any two positive numbers $a_0$ and $b_0$, the arithmetic-geometric-mean iterate is
$a_{i+1} = (a_i + b_i)/2$, $b_{i+1} = \sqrt{a_ib_i}$. For $a_0\gg b_0$ this converges rapidly to $a=\lim_{i\rightarrow\infty} a_i$.
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