Question

Find the exact value of each expression.

(a) \(\ln \frac{1}{{{e^2}}}\) (b) \(\ln \sqrt e \) (c) \(\ln \left( {\ln {e^{{e^{50}}}}} \right)\)

Short answer

a) The exact value of \(\ln \frac{1}{{{e^2}}}\) is \( - 2\).

b) The exact value of \(\ln \sqrt e \) is \(\frac{1}{2}\).

c) The exact value of \(\ln \left( {\ln {e^{{e^{50}}}}} \right)\) is \(50\).

Step-by-step solution

Properties of natural logarithm function

The properties of natural logarithm function are defined as:

1. \(\ln x = y \Leftrightarrow {e^y} = x\)
2. \(\ln \left( {{e^x}} \right) = x\,,{\rm{ }}x \in \mathbb{R}\)
3. \({e^{\ln x}} = x\,,{\rm{ }}x > 0\)
4. \(\ln e = 1\)
5. \({x^r} = {e^{r\ln x}}\)

Determine the exact value of the expression

a)

Evaluate the exact value of the \(\ln \frac{1}{{{e^2}}}\) as shown below:

\(\begin{aligned}\ln \frac{1}{{{e^2}}} &= \ln {e^{ - 2}}\,\,\,\left( {\ln e = 1} \right)\\ &= - 2\end{aligned}\)

Thus, the exact value of \(\ln \frac{1}{{{e^2}}}\) is \( - 2\).

b)

Evaluate the exact value of \(\ln \sqrt e \) as shown below:

\(\begin{aligned}\ln \sqrt e &= \ln {e^{\frac{1}{2}}}\,\,\,\,\,\,\left( {\ln e = 1} \right)\\ &= \frac{1}{2}\end{aligned}\)

Thus, the exact value of \(\ln \sqrt e \) is \(\frac{1}{2}\).

c)

Evaluate the exact value of \(\ln \left( {\ln {e^{{e^{50}}}}} \right)\) as shown below:

\(\begin{aligned}\ln \left( {\ln {e^{{e^{50}}}}} \right) &= \ln \left( {{e^{50}}} \right)\,\,\,\,\,\,\,\,\,\left( {\ln e = 1} \right)\\ &= 50\end{aligned}\)

Thus, the exact value of \(\ln \left( {\ln {e^{{e^{50}}}}} \right)\) is \(50\).