Question

Use the properties of logarithms to expand the logarithmic expression. $$ \ln \frac{1}{e} $$

Short answer

The expanded and simplified expression for \( \ln \frac{1}{e} \) is -1.

Step-by-step solution

Convert Quotient to Difference

First, the logarithm of a quotient can be converted into the difference of two logarithms. In this case, \(\ln \frac{1}{e}\) will become the difference \( \ln (1) - \ln (e) \).

Evaluate Logarithms

Next, evaluate each of these individual logarithms. The value of \( \ln (1) \) is 0 because any number raised to the power 0 gives 1. The value of \( \ln (e) \) is 1 because \( e \) raised to the power 1 gives \( e \). In other words, 0 - 1 = -1.