The age of an ancient tree trunk is estimated using radiocarbon dating. If the trunk has a C-14 decay rate that is 34% of what it is in living plants, how old is the trunk?

I'm assuming that you are saying that there is only `34%` of the total carbon-14 in the trunk left. If that's the case, then we have to do the following steps to solve the problem.

We have to use the half-lives of carbon to calculate the age, as well as the fraction remaining.

Know that, after `n` half-lives of the substance have passed, there is only going to be `100/(2^n)%` of the substance left.

(Source: https://en.wikipedia.org/wiki/Half-life)

So, we can setup the following equation:

`100/(2^n)%=34%`

Removing percent from both sides, we get

`100/(2^n)=34`

`100*2^-n=34`

From here, cross multiplication gives us

`2^-n=34/100=17/50`

Now, we have to use logarithms. We can take the natural log of both sides, which yields

`ln(2^-n)=ln(17/50)`

Using the power rule for logarithms, `log_c(a^b)=blog_c(a)`, so we get

`-nln(2)=ln(17/50)`

Now, we can divide by `ln(2)` in both sides to get

`-n=(ln(17/50))/(ln(2))`

`n=-(ln(17/50))/(ln(2))`

From here, we need to use a calculator to figure out the answer. Plugging this into a calculator, it gives us

`n~~1.56` to `3` significant figures

So, the carbon-14 in the trunk has elapsed `1.56` half-lives. From the start, the problem tells us that the half-life of carbon-14 is `5730` years, so the carbon in the tree trunk is

`5730*1.56=8938.8~~8939` years old

I hope that my explanation was clear! Feel free to ask me anything in the comments or send me a message.