Question
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Tutor-Verified Answer
Answer
\( n = -0.5 \)
Solution
To solve the equation:
\[
e^{-0.5} = e^{n}
\]
we can take advantage of the fact that the exponential function \( e^x \) is one-to-one. This means that if \( e^a = e^b \), then \( a = b \).
Applying this property to your equation:
\[
-0.5 = n
\]
**Solution:**
\[
n = -0.5
\]
So, \( n = -0.5 \).
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Simplify this solution The Deep Dive
To solve for \( n \) in the equation \( e^{-0.5} = e^n \), we can take the natural logarithm of both sides. Since the natural logarithm and the exponential function are inverses, we have: \[ \ln(e^{-0.5}) = \ln(e^n) \] This simplifies to: \[ -0.5 = n \] So, \( n = -0.5 \). Well done, you just tackled an exponential equation! Remember, the key to working with exponents and logarithms is to leverage their properties, particularly that they are inverses of each other. Keep practicing, and you'll become a pro at these kinds of problems!
