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Question: A scientist measures the initial amount of Carbon-14 in a substance to be 25 grams. The relationship between A , the amount of Carbon-14 remaining in that substance, in grams, and t , the elapsed time, in years, since the initial measurement is modeled by the following equation. A = 25e^0.00012t In how many years will the substance contain exactly 20 grams of Carbon-14 ? Give an exact answer expressed as a natural logarithm.

To solve for t, capital 'T' in this equation is what we want to find. We set A = 20 in your equation and solve for t: 20 = 25e^0.00012t First, we isolate the term with t by dividing both sides by 25: 20/25 = e^0.00012t or, 0.8 = e^0.00012t On this step, we will take the natural logarithm (ln) of both sides, because the natural logarithm and e are inverse functions, meaning taking the natural logarithm on the right side will remove exponentiation: ln(0.8) = ln(e^0.00012t) Due to a property of logarithms, we can then move the exponent in front of the natural logarithm on the right side: ln(0.8) = 0.00012t * ln(e) Since ln(e) equals 1: ln(0.8) = 0.00012t Finally, to solve for t, we will divide both sides by 0.00012: t = ln(0.8) / 0.00012 This is the exact answer.

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