Exponential Equations

An exponential equation is an equation in which a variable appears in the exponent. For example, $$y = a(b^x)$$ where a and b are constants.

Exponential growth occurs when a quantity increases by a consistent percentage over a period of time, represented by the equation $$y = a(1 + r)^t$$ .

Exponential decay occurs when a quantity decreases by a consistent percentage over time, represented by the equation $$y = a(1 - r)^t$$ .

The base of an exponential function is the constant factor that is raised to the power of the variable. In $$y = a(b^x)$$ , b is the base.

The y-intercept of an exponential function is the value of y when x = 0, calculated as $$y = a(b^0) = a$$ .

Exponential growth is identified by a curve that rises steeply to the right, indicating that the function increases rapidly as x increases.

Exponential decay is identified by a curve that falls steeply to the right, indicating that the function decreases rapidly as x increases.

The general form of an exponential function is $$y = a(b^x)$$ , where a is a constant, b is the base, and x is the exponent.

The constant 'a' represents the initial value or the y-intercept of the function when x = 0.

If the base 'b' is greater than 1, the function represents exponential growth.