Large Logs

The number of trialing zeroes are given by:

$\begin{aligned} x & = \left \lfloor \log{\left( 100^{1000^{10000}}\right)} \right \rfloor = \left \lfloor \log{\left( 10^{2\times 10^{3\times 10000}}\right)} \right \rfloor = 2 \times 10^{30000} \\ y & = \left \lfloor \log{\left( 1000^{10000^{100000}}\right)} \right \rfloor = \left \lfloor \log{\left( 10^{3\times 10^{4\times 100000}}\right)} \right \rfloor = 3 \times 10^{400000} \end{aligned}$

Now, we have:

$\begin{aligned} \left \lfloor \log_x{y} \right \rfloor & = \left \lfloor \dfrac {\log{y}} {\log{x}} \right \rfloor = \left \lfloor \dfrac {\log{\left(3 \times 10^{400000}\right)}} {\log{\left(2 \times 10^{30000}\right)}} \right \rfloor = \left \lfloor \dfrac {\log{3} + 400000} {\log{2} + 30000} \right \rfloor \\ & = \left \lfloor \dfrac {400000} {30000} \right \rfloor = \left \lfloor \dfrac {40} {3} \right \rfloor = \boxed{13} \end{aligned}$