#### Answer

6

#### Work Step by Step

$\int^e_1 \int^{e^2}_1 \int^{e^3}_1 \frac{1}{xyz}$ dx dy dz
=$\int^e_1 \int^{e^2}_1 [\frac{ln x}{yz}]^{e^3}_1 dy $ dz
=$\int^e_1 \int^{e^2}_1 \frac{3}{yz}$ dy dz
=$3 \int^e_1 [\frac{ln y}{z}]^{e^2}_1$ dz
=$\int^e_1 \frac{6}{z}$ dz
=6