Question
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Tutor-Verified Answer
Answer
\( x = \frac{1}{7e} \)
Solution
1. Start with the equation:
\[
2 \ln (7x) = -2
\]
2. Divide both sides by 2:
\[
\ln (7x) = -1
\]
3. Exponentiate both sides to remove the logarithm:
\[
e^{\ln (7x)} = e^{-1} \quad \Rightarrow \quad 7x = e^{-1}
\]
4. Solve for \( x \) by dividing both sides by 7:
\[
x = \frac{e^{-1}}{7} = \frac{1}{7e}
\]
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Simplify this solution Extra Insights
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: \[ x = \frac{1}{7e} \]
