Chapter 6.6, Problem 6E

To determine

To calculate: .The equation for the value of $x$ .

Answer to Problem 6E

The required value of $x$ is $1$ .

Explanation of Solution

Given Information:

The given equation is $e^{2x}=e^{3x-1}$ .

Formula Used Power rule of logarithmic functions:

  ${\log }_a{x}^p=p{\log }_{a}x$

Where $a$ is the base, $x$ is the variable and $p$ is the power of variable.

Calculation:

Consider the equation $e^{2x}=e^{3x-1}$ .

Take natural log with base $e$ of the above equation.

  ${\log }_e{e}^{2x}={\log }_e{e}^{3x-1}$

Now, use power rule of logarithm ${\log }_a{x}^p=p{\log }_{a}x$ in the above equation.

  $2x{\ln }_{e}e=\left(3x-1\right){\ln }_{e}e$

Substitute $1$ for ${\ln }_{e}e$ in the above equation.

  $\begin{array}{c}2x\left(1\right)=\left(3x-1\right)\left(1\right)\\ 2x=3x-1\\ 2x-3x=-1\\ -x=-1\end{array}$

Simplify the above equation $-x=-1$ .

  $\begin{array}{c}-x=-1\\ x=1\end{array}$

Hence, the required value of $x$ is $1$ .