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You can simply start with the definition of gross returns \begin{align*} R_{t+1}&=\frac{D_{t+1}+P_{t+1}}{P_t} \\ &=\frac{1+P_{t+1}/D_{t+1}}{P_t/D_t}\frac{D_{t+1}}{D_t}, \end{align*} where the first fraction contains now your price dividend ratio. Going to log-returns, \begin{align*} r_{t+1} &= \ln\left(1+\frac{P_{t+1}}{D_{t+1}}\right) - \ln\left(\...


The second expression is just another representation of the former and has nothing to do with continuous compounding. Instead note that $\log(a)-\log(b)=\log\left(\frac{a}{b}\right)$ from which the result should become immediately clear.

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