Unlimited growth: Exponential growth
Doubling time and half-life for exponential growth
Sometimes, when working with exponential growth, not the growth factor or relative growth rate is given, but the doubling time \(t_d\) or, in case of exponential decay, the half-life \(t_{1/2}\). The doubling time and half-life you can associate with the growth factor \(g\) and relative growth rate constant \(r\).
For exponential decay we have \(\displaystyle g^{t_{1/2}}=\frac{1}{2}\). By taking the natural logarithm on both sides and using the calculation rules for logarithms you get: \[\ln(g^{t_{1/2}})=\ln(\frac{1}{2})\] Thus \[t_{1/2} \cdot \ln(g) = -\ln(2)\] After rewriting: \[t_{1/2} = -\frac{\ln(2)}{\ln(g)}\] Because \(g=e^r\), we can also write \(\ln(g)=r\) and \[t_{1/2} = -\frac{\ln(2)}{r}\approx -\frac{0.693}{r}\]
For exponential growth holds \(g^{t_{d}}=2\). By taking the natural logarithm on both sides and using the calculation rules for logarithms you get: \[t_{d} = \frac{\ln(2)}{\ln(g)}\] Because \(g=e^r\), we can also write \[t_{d} = \frac{\ln(2)}{r}\approx \frac{0.693}{r}\]
