I'm trying to understand the formula to move from log points to percentage points. I know the same question has already been asked here: How to interpret the difference in log points
and I can follow PaulB's answer easily until the taylor expansion, is the last step that I have troubles understanding. Could anyone please help me claryfying that? It seems to me like the "-1" should be part of the log, but it's clearly not correct.
Any help would be highly appreciated
We have that when x is "small" (let try with calculator)
$$\log(1+x)\approx x$$
therefore for small pecentage $\frac{\%\Delta}{100}$
$$\log\left(1+\frac{\%\Delta}{100}\right) \approx \frac{\%\Delta}{100}$$
For the last step, by exponentiation
$$\log(x)-\text{log}(y)=\log\left(\frac{\%\Delta}{100}+1\right)$$
$$e^{\log(x)-\text{log}(y)}=e^{\log\left(\frac{\%\Delta}{100}+1\right)}=\frac{\%\Delta}{100}+1$$
$$\frac{\%\Delta}{100}=e^{\log(x)-\text{log}(y)}-1$$