Question

evaluate.
\\(\ln \frac{1}{e^{6}}\\)
\\(\ln \frac{1}{e^{6}} = \square\\)

Explanation:

Step1: Use logarithm property of reciprocals

Recall that $\ln\frac{1}{x}=-\ln x$. So, $\ln\frac{1}{e^{6}}=-\ln e^{6}$.

Step2: Use logarithm power rule

The power rule of logarithms states that $\ln a^{b}=b\ln a$. Here, $a = e$ and $b = 6$, so $\ln e^{6}=6\ln e$.

Step3: Evaluate $\ln e$

We know that $\ln e = 1$ (since the natural logarithm of its base $e$ is 1). Thus, $6\ln e = 6\times1 = 6$.

Step4: Combine results

From Step1, we have $-\ln e^{6}$, and from Step3, $\ln e^{6}=6$, so $-\ln e^{6}=-6$.

Answer:

-6