Given dtdB∝B⇒dtdB=λB,(λis proportionility constant)
⇒BdB=λdt
Integrate both sides with proper limit we get,
∫B1B2BdB=∫t1t2λdt
⇒lnB1B2=λ(t2−t1)...(i)
Given that, at t=0,B=1000&t=20%\text{increase}=1200
∴ln10001200=2λ⇒λ=21ln56
Now an equation (i)becomes,
lnB1B2=21ln56(t2−t1)...(ii)
Given that, at {t}_{2}=\frac{k}{\mathrm{ln}2},{B}_{2}=2000&t=0,{B}_{1}=1000
From (ii),ln10002000=21ln56(ln56k−0)=21k
⇒k=2ln2
∴(ln2k)2=(ln22ln2)2=4