# Logarithm-exponential Equation Conundrum

Calculus Level 5

Ahmad was trying to find the number of solutions to

$\log_{(1/16)}x=\left(\frac{1}{16}\right)^x.$

He noticed that these functions are inverses of each other. He drew a sketch of the function $$y = \log_{(1/16)}x$$ in red, and then flipped it about the line $$y=x$$ (shown in dotted green), to obtain a sketch of the function $$y=\left(\frac{1}{16}\right)^x$$ in blue.

The picture suggests that the equation has one real solution. But algebra suggests that there are two: $$x=\frac{1}{4}$$ and $$x=\frac{1}{2}$$. How many real solutions does this equation actually have?

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