Question

evaluate each of the following. give answers as integers or reduced fractions.
(a) \\(\ln(e^4) = \\) enter your answer
(b) \\(\ln\left(\frac{1}{e^6}\
ight) = \\) enter your answer

Explanation:

Step1: Use the property of logarithms $\ln(a^b)=b\ln(a)$

For part (a), $\ln(e^{4})$, here $a = e$ and $b = 4$. Since $\ln(e)=1$, we have $\ln(e^{4})=4\ln(e)$.

Step2: Substitute $\ln(e) = 1$

$4\ln(e)=4\times1 = 4$.

Step3: Rewrite $\frac{1}{e^{6}}$ as $e^{-6}$

For part (b), $\ln(\frac{1}{e^{6}})=\ln(e^{-6})$.

Step4: Apply the property $\ln(a^b)=b\ln(a)$ again

Here $a = e$ and $b=-6$, so $\ln(e^{-6})=-6\ln(e)$.

Step5: Substitute $\ln(e) = 1$

$-6\ln(e)=-6\times1=-6$.

Answer:

(a) $4$
(b) $-6$