For small $x$, one has $\ln(1+x)=x$? What does it mean that for small $x$, one has $\ln(1+x)=x$? How can you explain this thing ? Thanks in advance for your reply.

Take the tangent line at of $f(x) = \ln(1+x)$ in $x = 0$. 
\begin{align*}
f(x) & \approx f(0) + f'(0) (x - 0) \\
& = \ln(1+0) + \left[\frac{d}{dx} \ln(1+x)\right]_{x = 0} (x-0) \\
& = 0 + 1 x \\
& = x
\end{align*}
