Question

Let a curve \(y=f(x)\) pass through the point \((2,(\log_e2)^2)\) and have slope \(\frac{2y}{x\log_ex}\) for all positive real value of \(x\). Then the value of \(f(e)\) is equal to ______________.

Hint:

Separate variables: \(\frac{dy}{y}=\frac{2\,dx}{x\ln x}\).

Answer 1

Correct Answer: 1

\(\frac{dy}{dx}=\frac{2y}{x\ln x}\Rightarrow\frac{dy}{y}=\frac{2\,dx}{x\ln x}\Rightarrow\ln y=2\ln(\ln x)+C\Rightarrow y=K(\ln x)^2\).
At \(x=2\): \((\ln2)^2=K(\ln2)^2\Rightarrow K=1\).
So \(y=(\ln x)^2\) and \(f(e)=(\ln e)^2=1\).
Answer: 1