Question

Which equation is equivalent to 2^4x = 8^x-3?

Answer 1

Answer:  the correct option is

(D)  $2^{4x}=2^{3x-9}.$

Step-by-step explanation:  We are given to select the correct equation that is equivalent to the following equation :

$2^{4x}=8^{x-3}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

Equivalent equations means  two equations that can be obtained from one another using some properties of formula.

We will be using the following formula :

$(a^b)^c=a^{b\times c}.$

From equation (i), we have

$2^{4x}=8^{x-3}\\\\\Rightarrow 2^{4x}=(2^3)^{x-3}\\\\\Rightarrow 2^{4x}=2^{3\times(x-3)}\\\\\Rightarrow 2^{4x}=2^{3x-9}.$

Thus, the required equivalent equation is $2^{4x}=2^{3x-9}.$

Option (D) is CORRECT.

Answer 2

Answer:

$\large\boxed{2^{4x}=2^{3x-9}}$

Step-by-step explanation:

$8=2^3\to 8^{x-3}=(2^3)^{x-3}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\=2^{3(x-3)}\qquad\text{use the distributive property}\ a(b+c)=ab+ac\\\\=2^{(3)(x)+(3)(-3)}=2^{3x-9}$

If you want a solution of this equation:

$2^{4x}=8^{x-3}\\\\2^{4x}=2^{3x-9}\iff4x=x-3\qquad\text{subtract}\ x\ \text{from both sides}\\\\3x=-3\qquad\text{divide both sides by 3}\\\\x=-1$