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Solved on Jan 29, 2024

Find the natural logarithm of the reciprocal of the mathematical constant $e$ .

STEP 1

Assumptions
1. We need to evaluate the natural logarithm of the reciprocal of $e$ , where $e$ is the base of the natural logarithm.

STEP 2

Recall the definition of the natural logarithm. The natural logarithm of a number $x$ , denoted as $\ln(x)$ , is the power to which $e$ must be raised to obtain the number $x$ .

STEP 3

Recognize that the natural logarithm of $e$ itself is 1, because $e^1 = e$ . This is a fundamental property of logarithms.
$\ln(e) = 1$

STEP 4

Understand that the natural logarithm function is the inverse of the exponential function with base $e$ . Therefore, the logarithm of a reciprocal is the negative of the logarithm of the number.
$\ln\left(\frac{1}{x}\right) = -\ln(x)$

STEP 5

Apply the property from STEP_4 to the problem at hand, where $x = e$ .
$\ln\left(\frac{1}{e}\right) = -\ln(e)$

STEP 6

Substitute the value from STEP_3 into the equation from STEP_5.
$\ln\left(\frac{1}{e}\right) = -1$
The solution to the problem $\ln \frac{1}{e}$ is $-1$ .

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