Chapter 6.4, Problem 24E

To determine

To match the equation to the graph that can be used to solve it then solve it.

Answer to Problem 24E

Given equation matches with graph (A)

Solution to the given equation is $x=-4$

Explanation of Solution

Given information:

Given equation $2^{-x-2}=4$

Given graphs

  

Concept Used:

Intersection point given in the graph must satisfy the given equation $2^{-x-2}=4$

Given equation $2^{-x-2}=4$

Make table from given information:

Graph	Intersection point	Value of x-intercept	$2^{-x-2}$
(A)	(-4,4)	-4	${\left(2\right)}^{-\left(-4\right)-2}=2^{4-2}=2^2=4$
(B)	(0,4)	0	${\left(2\right)}^{-0-2}=2^{-2}=\frac{1}{4}$
(C)	(2,4)	2	${\left(2\right)}^{-2-2}=2^{-4}=\frac{1}{16}$
(D)	(3,4)	3	${\left(2\right)}^{-3-2}=2^{-5}=\frac{1}{32}$

Clearly, graph (B) satisfies the equation $2^{x+2}=4$

Also

  $\begin{array}{l}{\left(2\right)}^{-x-2}=2^2\\ \text{Since bases are equal powers must be equal}\\ \text{Therefore},\\ -x-2=2\\ -x=2+2\\ -x=4\\ x=-4\end{array}$

Hence,

Solution of given equation is $x=-4$

Conclusion:

Given equation matches with graph (A)

Solution to the given equation is $x=-4$