Given that the half-life of uranium is [tex]4.5\times10^9[/tex] years, you know that after that amount of time you're left with half as much uranium as what you start with:
[tex]\dfrac12=e^{-k\times4.5\times10^9}[/tex]
Solve this equation for [tex]k[/tex] to find the decay rate.
[tex]\dfrac12=e^{-k\times4.5\times10^9}[/tex]
[tex]\implies \ln\dfrac12=\ln e^{-k\times4.5\times10^9}[/tex]
[tex]\implies -\ln2=-k\times4.5\times10^9\ln e[/tex]
[tex]\implies k=\dfrac{\ln2}{4.5}\times10^{-9}[/tex]
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 [tex]t[/tex] such that
[tex]\dfrac18=e^{-kt}[/tex]
[tex]\implies\ln\dfrac18=\ln e^{-kt}[/tex]
[tex]\implies-\ln8=-kt\ln e[/tex]
[tex]\implies t=\dfrac{\ln8}k=\dfrac{\ln8}{\frac{\ln2}{4.5}\times10^{-9}}=\dfrac{\log_28}{4.5}\times10^9\approx0.66\times10^9[/tex]
