Introduction to Exponential Functions

Exponential functions are fundamental in modeling dynamic processes where the rate of change is proportional to the current value. Defined by f(x) = b^x, where b is a positive constant not equal to one, they form the basis for describing processes of growth and decay depending on whether the base b is greater than or less than one.

Exponential models describe situations where change occurs at a rate proportional to the current amount. These include phenomena such as bacterial proliferation, radioactive substance decay, and depreciation of assets. The general form of an exponential model can be expressed as

Here, A₀​ is the initial amount, b is the base indicating the growth (b > 1) or decay (0 < b < 1) factor, and t represents time or another continuous variable.

Transformations such as translations, reflections, and stretches alter the graphical representation of exponential functions but preserve their fundamental shape and asymptotic behavior. Additionally, the One-to-One Property of exponential functions ensures that if b^x = b^y, then x = y, allowing for solving equations involving exponents through base comparison.