Please take a look at the following snippet of a paper. According to the underlined part, eq(3)(4) can be obtained by taking logs of eq(1)(2), so by taking log of eq(1) (and ignore expectation) I get
$$\frac{1+r_{t}}{1+r_{t}^{i}}=\frac{S_{t+1}^{i}}{S_{t}^{i}}\Rightarrow$$ $$ln(\frac{1+r_{t}}{1+r_{t}^{i}})=ln(\frac{S_{t+1}^{i}}{S_{t}^{i}})=ln(S_{t+1}^{i})-ln(S_{t}^{i}) $$
However please help me understand how to convert $$ln(\frac{1+r_{t}}{1+r_{t}^{i}})$$ to $$r_{t}-r_{t}^{i}$$
Thanks a lot!
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