Labor demand and supply curves in terms of elasticities
This result is presented on page 14 of Foundations of Modern Macroeconomics. The key is to total differentiate both expressions.
Start with the supply of labor:
\begin{gather}
W/P^e = g(N^S) \\
d(W/P^e) = d\Big( g(N^S) \Big) \\
\frac{1}{P^e} dW - \frac{W}{(P^e)^2}dP^e = g_NdN^S \\
dN^S = \frac{1}{g_N}\frac{dW}{P^e} - \frac{1}{g_N}\frac{W}{P^e}\frac{dP^e}{P^e} \\
dN^S = \frac{1}{g_N}\frac{dW}{P^e} - \frac{g(N^S)}{g_N}\frac{dP^e}{P^e}, \quad \text{where} \quad W/P^e = g(N^S) \\
dN^S = \frac{1}{g_N}\frac{g(N^S)}{W}dW - \frac{g(N^S)}{g_N}\frac{dP^e}{P^e} \\
dN^S = \frac{g(N^S)}{g_N} \bigg[ \frac{dW}{W} - \frac{dP^e}{P^e} \bigg] \\
\frac{dN^S}{N^S} = \frac{g(N^S)}{N^Sg_N} \bigg[ \frac{dW}{W} - \frac{dP^e}{P^e} \bigg]
\end{gather}
To derive the expression for the demand of labor, one can follow a similar process (I will ommit the superscript $N^D$ for simplicity, but we know we are talking about labor demand). Recall that the FOC of the firm's problem is given by:
\begin{gather}
PF_N(N,\bar{K}) = W \\
F_N(N,\bar{K}) = \frac{W}{P} \\
d\Big( F_N(N,\bar{K}) \Big) = d\Big( W/P \Big) \\
F_{NN}dN + F_{NK}d\bar{K} = \frac{1}{P}dW - \frac{W}{P^2}dP \\
F_{NN}dN + F_{NK}d\bar{K} = F_N\frac{dW}{W} - F_N\frac{dP}{P} \\
F_{NN}dN = F_N \bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] - F_{NK}d\bar{K} \\
dN = -\frac{F_{NK}}{F_{NN}}d\bar{K} + \frac{F_N}{F_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] \\
\frac{dN^D}{N^D} = -\frac{F_{NK}}{N^DF_{NN}}d\bar{K} + \frac{F_N}{N^DF_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] \\
\frac{dN^D}{N^D} = \frac{d\bar{K}}{\bar{K}} + \frac{F_N}{N^DF_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg], \quad \text{where} \quad KF_{NK} = -NF_{NN}
\end{gather}