From the aggregate supply equation to the definition of inflation rate By definition the  inflation rate is $$\pi=\dfrac{P-P_{-1}}{P_{-1}}\cdot100\%$$ or could be defined in terms of the consumer price index CPI, but in this case I think the former is the one to consider.
From the book Macroeconomia by Blanchard, Amighini, and Giavazzi,
the aggregate supply equation is \begin{equation}
P=P^e(1+\mu)F(u^-,z^+)
\end{equation}
From here, and taking a specific $F:$
\begin{align}
    P&amp;=P^e(1+\mu)e^{-\alpha u+z} \nonumber\\
    \ln(P)&amp;=\ln(P^e)+\ln(1+\mu)-\alpha u+z  \nonumber\\
    \ln(P)-\ln(P_{-1})&amp;=\ln(P^e)-\ln(P_{-1})+\ln(1+\mu)-\alpha u+z \nonumber\\
     \ln(P)-\ln(P_{-1})&amp;=\ln(P^e)-\ln(P_{-1})+\mu-\alpha u+z\dots \text{due to $\ln(1+\mu)\approx\mu$ because $\mu$ is close to $0$ } \nonumber
  \end{align}
Thus \begin{align}\pi=\pi^e+(\mu+z)-\alpha u \end{align}
Why? For example how to pass from  $\ln(P)-\ln(P_{-1})$ to $\pi$?
We know that $\ln(P)-\ln(P_{-1})=\ln\dfrac{P}{P_{-1}}\fbox {=?}\dfrac{P}{P_{-1}}-1$
Thank you in advance
This is because for small values $x$, $$\ln x_{t+1} - \ln x_{t} \approx \frac{x_{t+1}-x_{t}}{x_{t}}.$$
This holds since the growth rate $g$ can be expressed as follows:
$$g= \frac{x_{t+1}-x_{t}}{x_{t}} \implies x_{t+1} = (1+g)x_{t}$$
taking logs we get that:
$$ \ln x_{t+1} = \ln (1+g)+ \ln x_{t} \implies \\ \ln x_{t+1} -\ln x_{t} = \ln (1+g) $$
Finally, for small values of $g$ we get that $\ln (1+g) \approx g$ thus we get:
$$\ln x_{t+1} -\ln x_{t} \approx g $$
By the same token for small values of inflation (and inflation typically will take values $\pi&lt;0.1$) it is completely reasonable to just define inflation as $\pi = \ln P_t - \ln P_{t-1}$ (although it would be more appropriate to use $\approx$).