Locus of Points in Two and Three Dimensions
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Rewrite 4^(x-1) as (2²)^(x-1) = 2^(2x-2). So equation: 2^(2x-2) - 2^x = 8. Let y = 2^x. Then 2^(2x) = y², and 2^(2x-2) = y²/4. Equation: y²/4 - y = 8 → multiply by 4: y² - 4y - 32 = 0 → (y-8)(y+4)=0 → y=8 (since y>0). Then 2^x = 8 → x=3.
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