We need to solve: 70 × (1.025)^w > 90

February 22, 2026

We need to solve: 70 × (1.025)^w > 90

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The culture doubles every 3 hours, so in 18 hours, it doubles 18 / 3 = <<18 / 3 = 6>>6 times.
Each doubling multiplies the population by 2, so after 6 doublings: 500 × 2⁶ = 500 × 64 = <<500 * 64 = 32000>>32,000 bacteria.
An AI model's accuracy improves by 2.5% each week, starting at 70%. After how many weeks will its accuracy exceed 90%?
Divide both sides by 70: (1.025)^w > 90 / 70 = <<90 / 70 = 1.2857>>1.2857
Take log of both sides: w × log(1.025) > log(1.2857)
w > log(1.2857) / log(1.025) ≈ 0.1096 / 0.0107 ≈ <<0.1096 / 0.0107 ≈ 10.23>>10.23

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