Chapter 7.3, Problem 81PPS

To determine

To solve the given equation.

Answer to Problem 81PPS

  $x=-5$

Explanation of Solution

Given:

Equation: $2^{4x}={32}^{x+1}$

Formula used:

Exponent rules:

  ${(a^b)}^c=a^{bc}$

  $a^{f(x)}=a^{g(x)},$ then $f(x)=g(x)$

Distributive property: $a(b+c)=a\cdot b+a\cdot c$

Calculation:

Consider the given equation,

  $\Rightarrow 2^{4x}={32}^{x+1}$

Here, 32 can be written as $2^5.$

Hence,

  $\begin{array}{c}$ 2^{4x}={(2^5)}^{x+1}$$ 2^{4x}=2^{5(}{}^{x+1)}[\because {(a^b)}^c=a^{bc}]$4x=5(x+1)[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Simplifying the above linear equation,

  $4x=5(x+1)$

Applying distributive property to the right-hand side,

  $\begin{array}{l}4x=\left(5\times x\right)+\left(5\times 1\right)\\ 4x=5x+5\end{array}$

Subtracting 5x on both sides and simplifying,

  $\begin{array}{c}4x-5x=5x+5-5x\\ -x=5\end{array}$

Dividing − 1 on both sides and simplifying,

  $\begin{array}{c}\frac{-x}{-1}=\frac{5}{-1}\\ x=-5\end{array}$

Conclusion:

Therefore, the solution to the given equation is $x=-5.$