What is the gross accounting relation of Cobb-Douglas function?

Question

We have Cobb-Douglas function like this $Y=AK^\alpha L^{1-\alpha}$, in one of the book, it deduce like this:

How can we get this formula? $$\frac{\Delta Y}Y = \frac{\Delta A}A+\alpha\frac{\Delta K}K+(1-\alpha)\frac{\Delta L}L$$

Answer 1

Take logs of both sides, i.e. $$\log Y=\log A+ a \log K +(1-a)\log L$$ This gives: $$\Delta\log Y = \Delta\log A + a \Delta\log K +(1-a) \Delta\log L$$

Then use that $\frac{d}{dx}\log x= 1/x$, which yields $\Delta\log x=\Delta x/x$. Apply that to each log-diff above.