Chapter 1.6, Problem 52E

(a)

To determine

To solve: The equation, $\ln \left(\ln x\right)=1$ for x.

Answer to Problem 52E

The value of $x=e^e$.

Explanation of Solution

The given equation is $\ln \left(\ln x\right)=1$.

Taking natural exponential on both sides, $e^{\ln \left(\ln x\right)}=e^1$

Use the law of logarithm, $e^{\ln x}=x,\forall x>0$ and obtain, $\ln x=e$.

Again taking natural exponential on both sides, $e^{\ln x}=e^e$.

Again apply the law of logarithm, $e^{\ln x}=x,\forall x>0$ and obtain, $x=e^e$.

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

(b)

To determine

To solve: The equation, $e^{ax}=C{e}^{bx},a\ne b$ for x.

Answer to Problem 52E

The values of $x=\frac{\ln c}{a-b}$.

Explanation of Solution

The given equation is $e^{ax}=C{e}^{bx},a\ne b$.

Simplify the given equation as follows.

$\begin{array}{l}{e}^{ax}=C{e}^{bx}\\ \frac{e^{ax}}{e^{bx}}=C\\ e^{\left(a-b\right)x}=C\end{array}$

Taking natural log on both sides, $\ln e^{\left(a-b\right)x}=\ln c$.

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

$\begin{array}{l}\left(a-b\right)x=\ln c\\ x=\frac{\ln c}{a-b}\end{array}$

Thus, the values of $x=\frac{\ln c}{a-b}$.