Solve for \( x \) \[ 2 \ln (7 x)=-2 \]

To solve the equation \( 2 \ln (7x) = -2 \), let's start by isolating the logarithm. We can do this by dividing both sides by 2: \[ \ln(7x) = -1 \] Next, we can eliminate the natural logarithm by exponentiating both sides. Remember, if \( \ln(a) = b \), then \( a = e^b \): \[ 7x = e^{-1} \] Now, to find \( x \), we divide both sides by 7: \[ x = \frac{e^{-1}}{7} \] To express \( e^{-1} \) in a simpler form, we can rewrite it as: \[ x = \frac{1}{7e} \] Thus, the solution for \( x \) is: \[ x = \frac{1}{7e} \]