To find the second derivative of the function \( y=\log\left(\frac{x^2}{e^2}\right) \) with respect to \( x \), we follow these steps:
Using the properties of logarithms, rewrite the function: \( y=\log(x^2)-\log(e^2) \). Since \( \log(e^2)=2 \), this simplifies to: \( y=\log(x^2)-2 \).
Apply the property of logarithms \( \log(a^b)=b\log(a) \): \( y=2\log(x)-2 \).
Differentiate \( y \) with respect to \( x \) to find \( \frac{dy}{dx} \): \( \frac{dy}{dx} = 2\cdot\frac{1}{x} = \frac{2}{x} \).
Differentiate again to find \( \frac{d^2y}{dx^2} \): \( \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{2}{x}\right) = 2\cdot\left(-\frac{1}{x^2}\right)=\frac{-2}{x^2} \).
Thus, the value of \( \frac{d^2y}{dx^2} \) is \( \frac{-2}{x^2} \).