Chapter 8.2, Problem 35PPS

To determine

To calculate: The value of x in the equation ${\left(\frac{1}{8}\right)}^{3x+4}={\left(\frac{1}{4}\right)}^{-2x+4}$ .

Answer to Problem 35PPS

The value of x in the equation ${\left(\frac{1}{8}\right)}^{3x+4}={\left(\frac{1}{4}\right)}^{-2x+4}$ is $-\frac{4}{13}$ .

Explanation of Solution

Given information:

The given equation is ${\left(\frac{1}{8}\right)}^{3x+4}={\left(\frac{1}{4}\right)}^{-2x+4}$ .

Formula used:

Property of equality of exponential functions:

If $b^x=b^y$ then $x=y$for all $b>0$ ,and $b\ne 1$ .

Calculation:

Consider the given equation,

  ${\left(\frac{1}{8}\right)}^{3x+4}={\left(\frac{1}{4}\right)}^{-2x+4}$

Since $\frac{1}{8}$ can be written as $2^{-3}$ and $\frac{1}{4}$ can be written as $2^{-2}$ .So, rewrite the above equation as like this,

  $\begin{array}{c}{\left(\frac{1}{8}\right)}^{3x+4}={\left(\frac{1}{4}\right)}^{-2x+4}\\ 2^{-3\left(3x+4\right)}=2^{-2\left(-2x+4\right)}\\ 2^{-9x-12}=2^{4x-8}\end{array}$

Now, apply the property of equality of exponential functions:

  $\begin{array}{c}-9x-12=4x-8\\ -9x-4x=-8+12\\ -13x=4\\ x=-\frac{4}{13}\end{array}$

Therefore, The value of x in the equation ${\left(\frac{1}{8}\right)}^{3x+4}={\left(\frac{1}{4}\right)}^{-2x+4}$ is $-\frac{4}{13}$ .