Exponential and Logarithmic Models

The exponential model

An exponential function has the form $f(x)=k\,a^x$ (or equivalently $f(x)=k\,e^{rx}$), where the variable is in the exponent. The defining feature is a constant percentage rate of change rather than a constant amount: each unit step multiplies the output by the same factor $a$. When $a>1$ the model grows; when $0<a<1$ it decays. The coefficient $k$ is the initial value $f(0)$.

Exponential growth appears in compound interest, populations, and viral spread; exponential decay appears in radioactivity, drug concentration, and depreciation.

Worked example (compound growth). \$2000 is invested at 4% annual interest compounded yearly. The balance after $t$ years is $$B(t)=2000(1.04)^t.$$ The base 1.04 encodes "increase by 4% each year." After 10 years, $B(10)=2000(1.04)^{10}