Question

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

2^4x = 2^2x-3
2^4x = 2^2x-6
2^4x = 2^3x-3
2^4x = 2^3x-9​

Answer 1

`\large\boxed{2^(4x)=2^(3x-9)}`

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`

Answer 2

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* c).`

From equation (i), we have

`2^(4x)=8^(x-3)\\\\\Rightarrow 2^(4x)=(2^3)^(x-3)\\\\\Rightarrow 2^(4x)=2^(3*(x-3))\\\\\Rightarrow 2^(4x)=2^(3x-9).`

Thus, the required equivalent equation is
`2^(4x)=2^(3x-9).`

Option (D) is CORRECT.