A radioactive nucleus A with a half-life T , decays into a nucleus...

Let ${N}_{A}$ and ${N}_{B}$ be the number of molecules of $A$ and $B$ after time $t$.

Also, after time $t$, $\dfrac{{N}_{B}}{{N}_{A}} = 0.3$(given).

Also, let ${N}_{0}$ be the total number of nucleus initial

After time $t$, ${N}_{A} + {N}_{B} = {N}_{0}$

${N}_{A} + 0.3{N}_{A} = {N}_{0}$

$∴ {N}_{A} = \dfrac{{N}_{0}}{1.3}$

Also, the rate of disintegration of $A$

${N}_{A} = {N}_{0} {e}^{- \lambdat}$

$⇒ \dfrac{{N}_{0}}{1.3} = {N}_{0} {e}^{- \lambdat} ; \dfrac{1}{1.3} = {e}^{- \lambdat} orlog( 1.3 ) = \lambdat$

$t = \dfrac{\log ( 1 .3 )}{\lambda}$

$∴ t = \dfrac{Tlog( 1 .3 )}{\log ( 2 )} = \dfrac{Tlog( 1 .3 )}{\log 2}$.