Half-life of a substance is found by the next equation:
[tex]N(t)\text{ = }N_0\cdot(\frac{1}{2})^{\frac{t}{t0.5}}[/tex]where:
N(t) is the quantity of the substance remaining
N0 is the initial quantity of the substance
t is the time elapsed
t0.5 is the half-life
Replacing with t = 42, N(t) = 0.25 and N0 = 2, we get:
[tex]\begin{gathered} 0.\text{25 = 2}\cdot0.5^{\frac{42}{t0.5}} \\ \frac{0.25}{2}=0.5^{\frac{42}{t0.5}} \\ 0.125\text{ }=0.5^{\frac{42}{t0.5}} \\ \ln (0.125)\text{ = }\frac{42}{t0.5}\cdot\ln (0.5) \\ t_{0.5}=\frac{42\cdot\ln (0.5)}{\ln (0.125)} \\ t_{0.5}=\text{14 days} \\ \end{gathered}[/tex]