Summary of Exponential Growth and Decay

Section 13.8 Summary of Exponential Growth and Decay

Table 13.8.1. Summary of Exponential Formulas

Formula 🔗	🔗
\(A=A_0 b^t\) 🔗	\(A \rightarrow\) amount after time \(t\) 🔗 \(A_0 \rightarrow\) initial amount 🔗 \(b \rightarrow\) growth factor (growth if \(b \gt 1\text{,}\) decay if \(0 \lt b \lt 1\)) 🔗
Percentage change 🔗 \(A=A_0(1+r)^t\) 🔗	\(A \rightarrow\) amount after \(t\) time periods 🔗 \(A_0 \rightarrow\) initial amount 🔗 \(r \rightarrow\) rate per period (as a decimal) 🔗 \(t \rightarrow\) number of periods 🔗
Half-life 🔗 \(A=A_0\brac{\frac{1}{2}}^{t/T}\) 🔗	\(T \rightarrow\) half-life (time to decay to \(\frac{1}{2}\)) 🔗
Growth over a time interval 🔗 \(A=A_0 b^{t/T}\) 🔗	\(b \rightarrow\) growth factor over \(T\) time 🔗 \(T \rightarrow\) time required to multiply the amount by \(b\) 🔗
Continuous growth 🔗 \(A=A_0 e^{kt}\) 🔗	\(e \approx 2.71\) (Euler’s number) 🔗 \(k \rightarrow\) continuous growth or decay rate 🔗

Decibel scale 🔗	🔗
\(\beta = 10\log\brac{\frac{I}{I_0}}\) 🔗	\(\beta \rightarrow\) sound level in decibels (dB) 🔗 \(I \rightarrow\) sound intensity (W/m\(^2\)) 🔗 \(I_0 \rightarrow\) reference intensity, \(10^{-12}\) W/m\(^2\) (threshold of hearing) 🔗
Richter scale 🔗	🔗
\(M = \log\brac{\frac{I}{I_0}}\) 🔗	\(M \rightarrow\) earthquake magnitude 🔗 \(I \rightarrow\) earthquake wave amplitude (microns) 🔗 \(I_0 \rightarrow\) reference amplitude (1 micron) 🔗
pH scale 🔗	🔗
\(\text{pH} = -\log\brac{[H^+]}\) 🔗	\(\text{pH} \rightarrow\) acidity or basicity of a solution 🔗 \([H^+] \rightarrow\) hydrogen ion concentration (mol/L) 🔗