Answer:
D. x = 3
Step-by-step explanation:
[tex]\frac{1}{2} ^{x-4}[/tex] - 3 = [tex]4^{x-3}[/tex] - 2
First, convert [tex]4^{x-3}[/tex] to base 2:
[tex]4^{x-3}[/tex] = [tex](2^{2})^{x-3}[/tex]
[tex]\frac{1}{2} ^{x-4}[/tex] - 3 = [tex](2^{2})^{x-3}[/tex] - 2
Next, convert [tex]\frac{1}{2} ^{x-4}[/tex] to base 2:
[tex]\frac{1}{2} ^{x-4}[/tex] = [tex](2^{-1})^{x-4}[/tex]
[tex](2^{-1})^{x-4}[/tex] - 3 = [tex](2^{2})^{x-3}[/tex] - 2
Apply exponent rule: [tex](a^{b})^{c}[/tex] = [tex]a^{bc}[/tex]:
[tex](2^{-1})^{x-4}[/tex] = [tex]2^{-1*(x-4)}[/tex]
[tex]2^{-1*(x-4)}[/tex] - 3 = [tex](2^{2})^{x-3}[/tex] - 2
Apply exponent rule: [tex](a^{b})^{c}[/tex] = [tex]a^{bc}[/tex]:
[tex](2^{2})^{x-3}[/tex] = [tex]2^{2(x-3)}[/tex]
[tex]2^{-1*(x-4)}[/tex] - 3 = [tex]2^{2(x-3)}[/tex] - 2
Apply exponent rule: [tex]a^{b+c}[/tex] = [tex]a^{b}[/tex][tex]a^{c}[/tex]:
[tex]2^{-1(x-4)}[/tex] = [tex]2^{-1x}[/tex] * [tex]2^{4}[/tex], [tex]2^{2(x-3)}[/tex] = [tex]2^{2x}[/tex] * [tex]2^{-6}[/tex]
[tex]2^{-1 * x}[/tex] * [tex]2^{4}[/tex] - 3 = [tex]2^{2x}[/tex] * [tex]2^{-6}[/tex] - 2
Apply exponent rule: [tex](a^{b})^{c}[/tex] = [tex]a^{bc}[/tex]:
[tex]2^{-1x}[/tex] = [tex](2^{x})^{-1}[/tex], [tex]2^{2x}[/tex] = [tex](2^{x})^{2}[/tex]
[tex](2^{x})^{-1}[/tex] * [tex]2^{4}[/tex] - 3 = [tex](2^{x})^{2}[/tex] * [tex]2^{-6}[/tex] - 2
Rewrite the equation with [tex]2^{x} = u[/tex]:
[tex](u)^{-1}[/tex] * [tex]2^{4}[/tex] - 3 = [tex](u)^{2}[/tex] * [tex]2^{-6}[/tex] - 2
Solve [tex]u^{-1}[/tex] * [tex]2^{4}[/tex] - 3 = [tex]u^{2}[/tex] * [tex]2^{-6}[/tex] - 2:
[tex]u^{-1}[/tex] * [tex]2^{4}[/tex] - 3 = [tex]u^{2}[/tex] * [tex]2^{-6}[/tex] - 2
Refine:
[tex]\frac{16}{u}[/tex] - 3 = [tex]\frac{1}{64}[/tex][tex]u^{2}[/tex] - 2
Add 3 to both sides:
[tex]\frac{16}{u}[/tex] - 3 + 3 = [tex]\frac{1}{64}[/tex][tex]u^{2}[/tex] - 2 + 3
Simplify:
[tex]\frac{16}{u}[/tex] = [tex]\frac{1}{64}[/tex][tex]u^{2}[/tex] + 1
Multiply by the Least Common Multiplier (64u):
[tex]\frac{16}{u}[/tex] * 64u = [tex]\frac{1}{64}[/tex][tex]u^{2}[/tex] + 1 * 64u
Simplify:
[tex]\frac{16}{u}[/tex] * 64u = [tex]\frac{1}{64}[/tex][tex]u^{2}[/tex] + 1 * 64u
Simplify [tex]\frac{16}{u}[/tex] * 64u:
1024
Simplify [tex]\frac{1}{64}[/tex][tex]u^{2}[/tex] * 64u:
[tex]u^{3}[/tex]
Substitute:
1024 = [tex]u^{3}[/tex] + 64u
Solve for u:
u = 8
Substitute back u = [tex]2^{x}[/tex]:
8 = [tex]2^{x}[/tex]
Solve for x:
x = 3
