GRE (3)

$\large \int_{e}^{a^e} \frac{dx}{x \int_{a}^{ax}\frac{dy}{y}} =1$

$\large \int_{e}^{a^e} \frac{dx}{x \ln y|_a^{ax }}=1$

$\large \int_{e}^{a^e} \frac{dx}{x \ln x} =1$

$\large \ln( \ln x)|_e^{a^e}=1$

$\large \implies \large \ln (\ln{a^e})= 1$

$\large\implies \ln a^e= e \implies \boxed {\large a=e }$

Similar solution with @Hana Nakkache 's

$\begin{aligned} I & = \int_e^{a^e} \frac {dx}{x\int_a^{ax} \frac {dy}y} = \int_e^{a^e} \frac {dx}{x\ln y\ \big|_a^{ax}} \\ & = \int_e^{a^e} \frac {dx}{x(\ln ax - \ln a)} = \int_e^{a^e} \frac {dx}{x\ln x} & \small \color{#3D99F6} \text{Let }u = \ln x \implies du = \frac {dx}x \\ & = \int_1^{e\ln a} \frac {du}u = \ln u\ \bigg|_1^{e\ln a} = \ln a \end{aligned}$

Therefore, $I=\ln a = 1$ $\implies a = \boxed{e}$ .