Chapter 6.6, Problem 68E

To determine

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

Answer to Problem 68E

The required value of $x$ is $2.88$ .

Explanation of Solution

Given Information:

The given equation is ${10}^{3x-8}=2^{5-x}$ .

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 is ${10}^{3x-8}=2^{5-x}$ .

Take common logarithm each side with base $10$ .

  ${\log }_{10}{10}^{3x-8}={\log }_{10}{2}^{5-x}$

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

  $\left(3x-8\right){\log }_{10}10=\left(5-x\right){\log }_{10}2$

Substitute $1$ for ${\log }_{10}10$ and $0.301$ for ${\log }_{10}2$ in the above equations.

  $\left(3x-8\right)\left(1\right)=\left(5-x\right)\left(0.301\right)$

Simplify the above equation.

  $\begin{array}{c}\left(3x-8\right)=\left(5-x\right)\left(0.301\right)\\ 3x-8=1.505-\left(0.301\right)x\\ 3x+\left(0.301\right)x=1.505+8\\ \left(3.301\right)x=9.505\end{array}$

Simplify the equation.

  $\begin{array}{c}\left(3.301\right)x=9.505\\ x=\frac{9.505}{3.301}\\ x=2.879\\ x\approx 2.88\end{array}$

Hence, the required value of $x$ is $2.88$ .