From the aggregate supply equation to the definition of inflation rate
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 $\approx$).