The problem requires us to evaluate the natural logarithm (ln) of several expressions involving the constant $e$. We need to evaluate the following: 1. $ln(\frac{1}{e^{12}})$

1. Problem Description

The problem requires us to evaluate the natural logarithm (ln) of several expressions involving the constant $e$. We need to evaluate the following:

1. $ln(\frac{1}{e^{12}})$

2. $ln(\sqrt[3]{e^8})$

3. $ln(e)$

4. $ln(\frac{1}{e})$

5. $ln(e^7)$

6. $ln(\sqrt[8]{e})$

7. $ln(\frac{1}{\sqrt{e^5}})$

2. Solution Steps

We will use the following properties of logarithms:

* $ln(e) = 1$

* $ln(e^x) = x$

* $ln(\frac{1}{x}) = -ln(x)$

* $ln(x^a) = a \cdot ln(x)$

* $\sqrt[n]{x} = x^{\frac{1}{n}}$

1. $ln(\frac{1}{e^{12}}) = ln(e^{-12}) = -12$

2. $ln(\sqrt[3]{e^8}) = ln((e^8)^{\frac{1}{3}}) = ln(e^{\frac{8}{3}}) = \frac{8}{3}$

3. $ln(e) = 1$

4. $ln(\frac{1}{e}) = ln(e^{-1}) = -1$

5. $ln(e^7) = 7$

6. $ln(\sqrt[8]{e}) = ln(e^{\frac{1}{8}}) = \frac{1}{8}$

7. $ln(\frac{1}{\sqrt{e^5}}) = ln(\frac{1}{e^{\frac{5}{2}}}) = ln(e^{-\frac{5}{2}}) = -\frac{5}{2}$

3. Final Answer

Here are the answers to the given expressions:

1. $ln(\frac{1}{e^{12}}) = -12$

2. $ln(\sqrt[3]{e^8}) = \frac{8}{3}$

3. $ln(e) = 1$

4. $ln(\frac{1}{e}) = -1$

5. $ln(e^7) = 7$

6. $ln(\sqrt[8]{e}) = \frac{1}{8}$

7. $ln(\frac{1}{\sqrt{e^5}}) = -\frac{5}{2}$