Suppose you find a rock that contains some potassium-40 (half-life of 1.3 billion years). you measure the amount and determine that there are 5 grams of potassium-40 in the rock. by measuring the amount of its decay product (argon-40) present in the rock, you realize that there must have been 40 grams of potassium-40 when the rock solidified. how old is the rock?

Equation for Half life:

A = a (0.5) ^ (t/h)

A is current amount, "a" is initial amount, h is halflife, t is time

5 = 40 (0.5) ^ (t/1.3x10^9)

5/40 = (0.5) ^ (t/1.3x10^9)

take the log of both sides, power rule

Log (5/40) = (t/1.3x10^9) * Log (0.5)

(1.3x10^9) * Log (5/40) / Log (0.5) = t

3.9x10^9 years = t

And if you think about what a half life is, the time it take for the amount to reduce to half.

40/2 = 20

20/2 = 10

10/2 = 5

It went through 3 half-lifes

3 * 1.3x10^9 = 3.9x10^9 years