A radioactive substance decays to $\frac{1}{8}$th of its initial activity in 30 days. The half life of the radioactive substance expressed in days is:

10

The correct answer is Option (2) → 10

Formula for Radioactive decay -

$A=A_0\left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$

where,

A, Remaining activity of substance

$A_0$, Initial activity

$t$, time elapsed = 30 days

$T_{1/2}$, half-life

$\frac{A}{A_0}=\frac{1}{8}$

$⇒\frac{1}{8}=\left(\frac{1}{2}\right)^{\frac{30}{T_{1/2}}}$

$⇒3ln2=\frac{30}{T_{1/2}}ln(2)$

$⇒T_{1/2}=10days$