Problem 92 - College Algebra Essentials

Question

Evaluate or simplify each expression without using a calculator. $$\ln \frac{1}{e^{7}}$$

Step-by-Step Solution

Answer

Simplified result for \( \ln \frac{1}{e^{7}} \) is -7.

Step 1: Understand The Meaning of Natural Logarithm

At first, let's recall that \(\ln x\), the natural logarithm, is the power to which \(\textit{e}\) would have to be raised to equal \(\textit{x}\). In other words, if: \(\textit{e}^y = x, then \ln x = y\). This property is fundamental for solving the given problem.

Step 2: Apply the Logarithm Properties

Next, apply logarithmic property: \(\ln{\frac{1}{a}} = -\ln{a} \). So, simplify the given expression as: \(\ln \frac{1}{e^{7}} = -\ln e^{7}\). However, applying the property of logarithms that says \(\ln{x^n} = n \ln x\), simplify to: \(-\ln e^{7} = -7 \ln e\).

Step 3: Simplify the Logarithm

Finally, \(\ln e\) equals 1 because it refers to the power to which \(\textit{e}\) raises to get \(\textit{e}\), which is obviously 1. Thus, -7 times 1 results into -7. So, \(-7 \ln e = -7\).