# 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?