Uranium-238 decays (through a long series of steps) to lead-206, with an overall effective half-life of 4.47 billion years, used in geological dating of very old rock samples. A rock sample that trapped no lead when it crystallized now contains U-238 and Pb-206 in a 4:3 mole ratio. Determine the rock's age, correctly reconstructing the INITIAL amount of U-238 before using the half-life formula, since the ratio here is not a clean power-of-two fraction.

Since every Pb-206 atom present came from decayed U-238, the ORIGINAL U-238 amount equals today's remaining U-238 plus today's Pb-206: with a 4:3 mole ratio of U-238 remaining to Pb-206 produced, the original U-238 total is 4+3=7 parts, of which 4 parts remain today. The surviving fraction is 4/7 ≈ 0.5714 — NOT a clean power of two like 1/2, 1/4, or 1/8, so the number of half-lives must be found using logarithms: n = log(4/7) ÷ log(0.5) ≈ 0.807 half-lives. Age = 0.807 × 4.47 ≈ 3.61 billion years, a realistic age for a very old terrestrial rock sample. Reading the ratio's second number (3) directly as a count of 3 half-lives (3 × 4.47 = 13.41 billion years) misreads a mole-ratio number as if it were already a half-life count. Reading the reconstructed total (7) as the half-life count instead (7 × 4.47 = 31.29 billion years) makes the same kind of error using the wrong number from the calculation. Assuming only 1 whole half-life has passed regardless of the actual ratio (reporting 4.47 billion years flat) skips the log-based calculation the non-power-of-two ratio actually requires.