Problem 91 - College Algebra Essentials

Question

Evaluate or simplify each expression without using a calculator. $$\ln \frac{1}{e^{6}}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-6\).

Step 1: Identify the logarithmic identity

Here we're working with a natural logarithm of a fraction. Looking at the expression, we can see that the base \(e\) in the denominator is being raised to the power \(6\). The logarithm of a fraction can be expressed as the difference of the logarithms of the numerator and the denominator. However, we specifically want to use the logarithmic identity \(ln(a^{-b}) = -b * ln(a)\), which would directly give us the answer.

Step 2: Apply the logarithmic identity

Applying the logarithmic identity to \(\ln \frac{1}{e^{6}}\), consider \(a^{-b}\) to be \(e^{6}\) (where \(a\) is \(e\) and \(b\) is \(6\)). The newly transformed version of the expression is \(-6*ln(e)\)

Step 3: Simplify the expression

The natural logarithm of \(e\) is \(1\). So, \(-6 * \ln(e)\) simplifies to \(-6 * 1\) which equals \(-6\).