A radioactive nucleus A with a half-life T, decays into a nucleus B. At t=0, there is no nucleus B. At some time t, the ratio of the number of B to that of A is 0.3. Then, t is given by:

Since the radioactive nucleus has half life 'T', ⇒λ=ln2T
A → B
At t=0, No. of A=N₀ and No. of B=0
Therefore at time=t,
No.of A=N₀e⁻λt
No.of B=N₀−N₀e⁻λt=N₀(1−e⁻λt)
⇒No.of BNo.of A=N₀(1−e⁻λt)N₀e⁻λt=0.3
⇒e⁻λt = 0.3 ⇒eλt=1.3
⇒λt=ln1.3
⇒t=ln1.3λ=Tln1.3ln2=Tlog1.3log2
Hence Correct answer is option (C).