Solved on Jan 29, 2024

Find the natural logarithm of the reciprocal of the mathematical constant ee.

STEP 1

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

STEP 2

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

STEP 3

Recognize that the natural logarithm of ee itself is 1, because e1=ee^1 = e. This is a fundamental property of logarithms.
ln(e)=1\ln(e) = 1

STEP 4

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

STEP 5

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

STEP 6

Substitute the value from STEP_3 into the equation from STEP_5.
ln(1e)=1\ln\left(\frac{1}{e}\right) = -1
The solution to the problem ln1e\ln \frac{1}{e} is 1-1.

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