Taylorseries , changing parameter to get another result

Question

I know Taylorseries of $ ln(1+x)= x -\frac{x}{2}+ \frac{x}{3}.... $, If i chnge $ln(1+x)$ to $ln(1+x^{2})$ , is it right to replace all X,s with $x^{2}$ ? I see people change sometime only one parameter to get result but i don't know exactly how they do it.