Usually, people write $y_t^{(n)}=-\frac{p_t^{(n)}}{n}$ where $y, p$ and log yield and log price respectively. My question is how do one derive this expression?
Note that $e^{-Y_t^{(n)}\cdot n}=P_t^{(n)}$ if $Y,P$ are the continuous yield and price of the one dollar $n$ period bond respectively at time $t$. Now, if we take logs, we get $$-Y_t^{(n)}\cdot n=p_t^{(n)}.$$ Therefore, we have $Y_t^{(n)}=-\frac{p_t^{(n)}}{n}$ which differs from the equation from the first paragraph. What went wrong?