Given that the half-life of uranium is \(4.5\times10^9\) years, you know that after that amount of time you're left with half as much uranium as what you start with:
\(\dfrac12=e^{-k\times4.5\times10^9}\)
Solve this equation for \(k\) to find the decay rate.
\(\dfrac12=e^{-k\times4.5\times10^9}\)
\(\implies \ln\dfrac12=\ln e^{-k\times4.5\times10^9}\)
\(\implies -\ln2=-k\times4.5\times10^9\ln e\)
\(\implies k=\dfrac{\ln2}{4.5}\times10^{-9}\)
Now you're looking to find the time it takes for the substance to decay to one eighth of its original amount, which is the time \(t\) such that
\(\dfrac18=e^{-kt}\)
\(\implies\ln\dfrac18=\ln e^{-kt}\)
\(\implies-\ln8=-kt\ln e\)
\(\implies t=\dfrac{\ln8}k=\dfrac{\ln8}{\frac{\ln2}{4.5}\times10^{-9}}=\dfrac{\log_28}{4.5}\times10^9\approx0.66\times10^9\)
