Linear approximation of a system described by a logarithm - how?
Functions like $\sqrt{\cdot}$ or $\log{(1+\cdot)}$ may emulate nonlinear system behavior like saturation. They are although used as companding methods, such as the $\mu$-law (or the A-law):
$$ x\to \mathrm{sgn}x\frac{\log{(1+\mu|x|})}{\log{(1+\mu)}}\,. $$
This can be used prior to quantization for signals with high-dynamics, and allows one to use uniform quantization and maintain a similar amount of relative errors for weak as well as strong signals (see for instance Digital filtering using logarithmic arithmetic (I glad to discover this work by Nick Kingsbury).
For telephony, people developed segment (&quot;piecewise linear&quot;, but actually piecewise affine to be correct) companding versions of such laws, for instance H. Kaneko, &quot;A Unified Formulation of Segment Companding Laws and Synthesis of Codecs and Digital Compandors&quot;. Basically, $\log(x+y)$ can be approximated by some affine law in its proper interval. I am aware of such approximations  for square roots: if $a\ge b$, then for instance with about 4% precision:
$$\sqrt{a^2+b^2} \sim 0.96a+0.4b  $$
yet I have to check for logarithms. In the same framework, as already said by @ben, if $a$ is a fixed quantity:

if $a&gt;b$, then $\log(a+b)\sim b/a+\log a$

which is  an affine approximation.