Question

Solve for x: ( 1/2 )^(x−1)=2^(3x−4)

Answer 1

`\huge\boxed{x=(5)/(4)}`

Explanation:

`\left((1)/(2)\right)^(x-1)=2^(3x-4)\qquad\text{use}\ a^(-1)=(1)/(a)\\\\\left(2^(-1)\right)^(x-1)=2^(3x-4)\qquad\text{use}\ (a^n)^m=a^(nm)\\\\2^((-1)(x-1))=2^(3x-4)\qquad\text{use the distributive property}:\ a(b+c)=ab+ac\\\\2^((-1)(x)+(-1)(-1))=2^(3x-4)\\\\2^(-x+1)=2^(3x-4)\iff-x+1=3x-4\qquad\text{subtract 1 from both sides}\\\\-x+1-1=3x-4-1\\\\-x=3x-5\qquad\text{subtract}\ 3x\ \text{from both sides}\\\\-x-3x=3x-3x-5\\\\-4x=-5\qquad\text{divide both sides by (-4)}`

`(-4x)/(-4)=(-5)/(-4)\\\\x=(5)/(4)`