Need a math help for the Cagan's model in macroeconomics
The answer to both your questions is that for small $x$ values
$$
\ln(1+x) \approx x,
$$
the difference being less than $x^2/2$. (Proof by Taylor-approximation.)

So if inflation is around 10%, then the absolute error from this type of approximation is less then 0.5%, which is pretty good.

This should also answer your second question, as the approximation
$$
\gamma x \approx \ln(1+ \gamma x),
$$
works as well.

It may also be worthwhile to look into elasticity.