Problem 72 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result of the expression \(\ln \frac{1}{e^{7}}\) is -7.

Step 1: Identification

Identify the number \(\frac{1}{e^{7}}\) as \(e^{-7}\). This because, taking any number to the negative power is equivalent to taking the reciprocal of that number to that power.

Step 2: Apply log law

Applying the logarithm law \(\ln(a^{-n}) = -n\ln(a)\), we substitute \(-7\) for \(n\) and \(e\) for \(a\). Hence, \(\ln \frac{1}{e^{7}} = \ln (e^{-7}) = -7\ln(e)\).

Step 3: Simplify

\(\ln(e)\) is the logarithm base \(e\) of \(e\), which is equal to 1, hence, \(-7\ln(e) = -7*1 = -7\).