Please Help! Thank You!

[math-a-phobic]

Hi and thank you for your time and assistance. I tried to solve these two math problems but got lost.

Problem 1:
Isotope C^14, Half-Life in Years 5730, and amount after 10,000 years 2g. Solve for the initial quantity and amount after 1000.

I tried using the equation Q=Ie^(rt) but got lost while substituting values. :?


Problem 2:
ln(x+1) - ln(x-2)= lnx^2
Solve for x.

I combined some terms due to the properties of logs and then got:
ln((x+1)/(x-2)) = lnx^2

Because I am unable to isolate the x, I think there is no solution to this problem. :x

Please help with these problems. Your help will be greatly appreciated. Thank you!

 

[galactus]

For the first problem, the decay constant is given by:

\(\displaystyle \L\\\frac{-1}{T}ln(2)=\frac{-1}{5730}ln(2)=-.000121\)

Set up the equation and solve for \(\displaystyle y_{0}\)

\(\displaystyle \L\\2=y_{0}e^{-.000121(10000)}\)

After you find \(\displaystyle y_{0}\), you can use that to find the amount after 1000 years.


For the second problem, you're on the right track.

\(\displaystyle \L\\ln(x+1)-ln(x-1)=ln{\frac{x+1}{x-1}}\)

You can take e to both sides and eliminate the ln's.

Now, solve \(\displaystyle \L\\\frac{x+1}{x-1}=x^{2}\) for x.