Chapter 12, Problem 3CR

To determine

To find: The solution set of the equation.

Answer to Problem 3CR

The solution set is n $\left\{\ln \left(\frac{5}{2}\right)\right\}$ .

Explanation of Solution

Given information:

The given equation is $2e^x=5$ .

Calculation:

Divide both of the equation by $2$ .

  $e^x=\frac{5}{2}$

Take natural logarithm on both sides using the property of logarithms.

  $\ln e^x=\ln \left(\frac{5}{2}\right)$

Rewrite the left side using the property of logarithms, ${\log }_a{M}^r=r{\log }_{a}M$ .

  $x\ln e=\ln \left(\frac{5}{2}\right)$

  $\ln e$ Is equivalent to ${\ln }_{e}e.$

  ${\ln }_{e}e=1$

Replace $\ln e$ with $1$ in the equation and simplify.

  $\begin{array}{l}x\left(1\right)=\ln \left(\frac{5}{2}\right)\\ x=\ln \left(\frac{5}{2}\right)\end{array}$

Therefore, the solution set is $\left\{\ln \left(\frac{5}{2}\right)\right\}$ ,

Therefore, the first five terms of the sequence are $0,\frac{3}{10},\frac{8}{11},\frac{5}{4}$ and $\frac{24}{13}$ .