Logarithm-exponential Equation Conundrum

Calculus Level 5

Ahmad was trying to find the number of solutions to

log(1/16)x=(116)x.\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 y = \log_{(1/16)}x in red, and then flipped it about the line y=xy=x (shown in dotted green), to obtain a sketch of the function y=(116)x 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=14x=\frac{1}{4} and x=12x=\frac{1}{2}. How many real solutions does this equation actually have?

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