Given a rock that contains $10^{20}$ atoms of a particular substance. Each atom has an exponentially distributed lifetime with a half-life of one century. How many centuries must pass before:
a) It is most likely 100 atoms remain
Textbook Answer:
Shouldn't $\lambda$ be:
?
Edit 1 (question from textbook):
Given the way the question is phrased, I wouldn't bother with $e$ or with $\lambda.$ Those are needed when you talk about instantaneous rates of change, but the question doesn't require that. \begin{align} & 10^{20} \times \left( \frac 1 2 \right)^\text{number of centuries} = 100 \\[15pt] & \left( \frac 1 2 \right)^\text{number of centuries} = \frac{100}{10^{20}} = 10^{-18} \\[15pt] & 2^\text{number of centuries} = 10^{18} \\[15pt] & \text{number of centuries} = \log_2 (10^{18}) = 18 \log_2 10. \end{align}
Your solution also appears to be correct. You have $\lambda = -\log \frac 1 2$ (where $\log$ means $\ln$ or $\log_e,$ as in your textbook), so \begin{align} 100 & = 10^{20} e^{\left(\ln\frac 1 2\right) t} = 10^{20} \left( \frac 1 2 \right)^t \end{align}