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Equation & Roots


How many roots does the equation 2^x=x^2 have?
Solution: For x > 0 and take logarithm on both sides. We get x \ln 2=2 \ln x which is equivalent to \frac{\ln x}{x}=\frac{\ln 2}{2}, which has at most 2 solutions, since the function is increasing on (0,e) and decreasing on (e,\infty).
On the negative side there is exactly one solution.

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