# Rate of change in a logarithmic model

Given the following model:

$$\ln{\left(W\right)}=\beta_0+\beta_1e+\beta_2e\ast x+\beta_3e\ast\ln{\left(P\right)} +\beta_4\ln{\left(P\right)}\ast x+ error.$$

Setting $$x = 4$$, and $$e =3$$, a $$1$$% increase in $$P$$ will lead to a __________ % increase in $$W$$?

I took the derivative with respect to $$P$$ and got:

$$\frac{\beta_3e}{P}+ \frac{\beta_4x}{P}$$

But don't know how to proceed.

You must also take the derivative of the right hand side: $$\frac{dW}{dP} \frac{1}{W} = \frac{\beta_3e}{P}+ \frac{\beta_4x}{P}$$ so with $$\delta W$$ the variation of $$W$$ you obtain: $$\frac{\delta W} {W} \simeq (\beta_3 e+\beta_4 x) \frac{\delta P}{P}$$
Note that you can obtain directly the result noticing that: $$d \ln(W) =\frac{dW}{W}, d \ln(P) =\frac{dP}{P}$$