# Mike's square

Geometry Level 5

Square $$ABCD$$ lies in the first quadrant, has area $$\frac{5}{64}$$, and side $$AB$$ is parallel to the $$x$$-axis. Vertices $$A, B$$, and $$C$$ lie on the graphs of $$y = \log_{a}x, y = 3\log_{a}x,$$ and $$y = 5 \log_{a}x,$$ respectively. The value of $$a^{ \frac{ \sqrt{5} } { 4} }$$ can be expressed as $$\frac{b}{c}$$ where $$b$$ and $$c$$ are positive coprime integers. What is the value of $$b+c$$?

This problem is shared by Michael T.

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