How to solve log word problem

Question

A superhero is rendered powerless when exposed to 70 or more grams of a certain element. A 650 year old rock that originally contained 350 grams of this element was recently stolen from a rock museum by the superhero’s nemesis. The half life of the element is known to be 300 years. How many grams of the element are still contained in the stolen rock? For how many years can this rock be used by the superhero’s nemesis to render the superhero powerless?

Answer 1

m(t) = m₀ (0.5)^t/300

 

m = mass in grams

t = years

m₀ = initial mass = 350 g

 

m(t) = 350 (0.5)^t/300

 

t = 650 years ⇒

 

m(650) = 350 (0.5)^650/300 ≅ 77.95 g

 

Reset present moment to t = 0, which establishes a new m₀ of 77.95 g, and find number of years T required to bring 77.95 g down to 70 g.

 

70 = 77.95 (0.5)^T/300

 

70/77.95 = (0.5)^T/300

 

log(70/77.5) = (T/300) log(0.5)

 

T = 300 log(70/77.95) / log(0.5) ≅ 46.56 years

 

You will be down to 70 g at 46.56 years after the time you had 77.95 g.