Chapter 1, Problem 26RE

(a)

To determine

To solve: The equation, $e^x=5$ for x.

Answer to Problem 26RE

Solution:

The value of $x=\ln 5$.

Explanation of Solution

The given equation is $e^x=5$.

Take the natural logarithm on both the sides, $\ln \left(e^x\right)=\ln 5$.

Use the law of logarithm, $\ln \left(e^x\right)=x,\forall x\in ℝ$ express the above equation as $x=\ln 5$.

Therefore, the value of $x=\ln 5$.

(b)

To determine

To solve: The equation, $\ln x=2$ for x.

Answer to Problem 26RE

Solution:

The value of $x=e^2$.

Explanation of Solution

The given equation is, $\ln x=2$.

Take the natural logarithm on both the sides, $e^{\ln x}=e^2$.

Use the law of logarithm $e^{\ln x}=x,\forall x>0$, it can be simplified as, $x=e^2$.

Thus, the value of $x=e^2$.

(c)

To determine

To solve: The equation, $e^{e^x}=2$ for x.

Answer to Problem 26RE

Solution:

The value of $x=\ln \left(\ln 2\right)$.

Explanation of Solution

The given equation is $e^{e^x}=2$.

Take the natural logarithm on both the sides, $\ln \left(e^{e^x}\right)=\ln 2$

Use the law of logarithm, $\ln \left(e^x\right)=x,\forall x\in ℝ$ twice and simplify as follows,

$\begin{array}{l}\ln \left(e^{e^x}\right)=\ln 2\\ e^x=\ln 2\\ \ln \left(e^x\right)=\ln \left(\ln 2\right)\\ x=\ln \left(\ln 2\right)\end{array}$.

Therefore, the value of $x=\ln \left(\ln 2\right)$.

(d)

To determine

To solve: The equation, ${\tan }^{-1}x=1$ for x.

Answer to Problem 26RE

Solution:

The value of $x=\tan 1$.

Explanation of Solution

The given equation is, ${\tan }^{-1}x=1$.

Express the given equation as, $x=\tan 1$

Therefore, the value of $x=\tan 1$.