The number of distinct real solutions of $\sin(\pi x) = \ln x$ is
Step-by-Step Solution
Key Concept: Plot $y = \sin(\pi x)$ (oscillates between $-1$ and $1$, period $2$) and $y = \ln x$ (increasing, passes through $0$ at $x=1$, approaches $-\infty$ as $x\to 0^+$). Count intersections by checking which oscillation cycles of $\sin(\pi x)$ cross $\ln x$.
Graphical analysis: $\sin(\pi x)$ completes full cycles in $[0,2], [2,4], \ldots$ For $x \in (0,1)$: $\ln x \in (-\infty, 0)$ crosses the negative half of the sine — 1 solution. For $x \in [1,e]$: $\ln x \in [0,1]$ crosses ascending/descending parts — multiple solutions. Total intersections by careful graphing: $\mathbf{6}$ solutions.
Correct Answer: 6