The point where the tangent to $f(x) = \log_2 x$ passes through $(0,0)$ is $x = e$ and the tangent is $x/(e\ln 2)$. So the equation $\log_2 x = cx$ has: - One solution for $c \leq 0$ - Two solutions for $c \in (0, 1/(e\ln 2))$ - One solution for $c = 1/(e\ln 2)$ - No solution for $c > 1/(e\ln 2)$
Hence the answer: $c \in (-\infty, 0] \cup \{1/(e\ln 2)\}$.