Step-by-step walkthrough:
a.
Well a standard half-life equation looks like this.
![N = N_0 * (\frac{1}{2})^{t/p](https://tex.z-dn.net/?f=N%20%3D%20N_0%20%2A%20%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7Bt%2Fp)
is the starting amount of parent element.
is the end amount of parent element
is the time elapsed
is a half-life decay period
We know that the starting amount is 74g, and the period for a half-life is 2.8 days.
Therefore you can create a function based off of the original equation, just sub in the values you already know.
![N(t) = 74g * (\frac{1}{2})^{t/2.8days](https://tex.z-dn.net/?f=N%28t%29%20%3D%2074g%20%2A%20%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7Bt%2F2.8days)
b.
This is easy now that we have already made the function. Here we just reuse it, but plug in 2.8 days.
![N(t) = 74g * (\frac{1}{2})^{t/2.8days} = N(2.8days) = 74g * (\frac{1}{2})^{2.8days/2.8days}\\= 74g * \frac{1}{2} = 37g](https://tex.z-dn.net/?f=N%28t%29%20%3D%2074g%20%2A%20%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7Bt%2F2.8days%7D%20%3D%20N%282.8days%29%20%3D%2074g%20%2A%20%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7B2.8days%2F2.8days%7D%5C%5C%3D%2074g%20%2A%20%5Cfrac%7B1%7D%7B2%7D%20%20%3D%20%2037g)
c.
Now we just gotta do some algebra. Use the original function but this time, replace
with 10g and solve algebraically.
![10g = 74g * (\frac{1}{2})^{t/2.8days}\\\\\frac{10g}{74g} = (\frac{1}{2})^{t/2.8days}](https://tex.z-dn.net/?f=10g%20%3D%2074g%20%2A%20%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7Bt%2F2.8days%7D%5C%5C%5C%5C%5Cfrac%7B10g%7D%7B74g%7D%20%3D%20%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7Bt%2F2.8days%7D)
Take the log of both sides.
![log(\frac{5}{37}) = log((\frac{1}{2})^{t/2.8days})](https://tex.z-dn.net/?f=log%28%5Cfrac%7B5%7D%7B37%7D%29%20%3D%20log%28%28%5Cfrac%7B1%7D%7B2%7D%29%5E%7Bt%2F2.8days%7D%29)
Use the exponent rule for log laws that, ![log(b^x) = x*log(b)](https://tex.z-dn.net/?f=log%28b%5Ex%29%20%3D%20x%2Alog%28b%29)
![log(\frac{5}{37}) = \frac{t}{2.8days} * log(\frac{1}{2})](https://tex.z-dn.net/?f=log%28%5Cfrac%7B5%7D%7B37%7D%29%20%3D%20%5Cfrac%7Bt%7D%7B2.8days%7D%20%2A%20log%28%5Cfrac%7B1%7D%7B2%7D%29)
![\frac{log(\frac{5}{37})}{log(\frac{1}{2})} = \frac{t}{2.8days}](https://tex.z-dn.net/?f=%5Cfrac%7Blog%28%5Cfrac%7B5%7D%7B37%7D%29%7D%7Blog%28%5Cfrac%7B1%7D%7B2%7D%29%7D%20%20%3D%20%5Cfrac%7Bt%7D%7B2.8days%7D)
![2.8 * \frac{log(\frac{5}{37})}{log(\frac{1}{2})} = t](https://tex.z-dn.net/?f=2.8%20%2A%20%5Cfrac%7Blog%28%5Cfrac%7B5%7D%7B37%7D%29%7D%7Blog%28%5Cfrac%7B1%7D%7B2%7D%29%7D%20%20%3D%20t)
slap that in your calculator and you get
![t = 8.1 days](https://tex.z-dn.net/?f=t%20%3D%208.1%20days)