Content
Exp & Log functions
Graphing exponential functions, exponential growth and decay, logarithmic inverse functions.
Graphing exponential functions, exponential growth and decay, logarithmic inverse functions.
No exercises available for this concept.
An exponential function has the form f(x)=ax for some base a>0 (and a=1). The domain of f is R, and the range is f(x)>0:
Powered by Desmos
In general, to graph an exponential function of the form f(x)=cax+k, find the y-intercept of the curve, then analyze the behavior of the function on both ends (as x→∞ and as x→−∞). If possible, plotting other easily calculated points - often f(1) or f(−1).
The y-intercept is at (0,c+k) because f(0)=ca0+k=c(1)+k.
On one end, the curve will approach y=k.
For a<1, as x→∞, f(x)→c(0)+k.
For a>1, as x→−∞, f(x)→c(0)+k.
On the other end, the curve will rise with increasing steepness.
Powered by Desmos
Exponential growth describes quantities that increase by the same factor over a certain amount of time. Algebraically, exponential growth is modeled by functions of the form
where b>1. b is called the growth factor.
Note: Aekt is another model for exponential growth if the instantaneous growth rate, k, is positive.
Stewart EJ, Madden R, Paul G, Taddei F (2005), CC BY-SA 4.0
Exponential decay describes quantities that decrease by the same factor over a certain amount of time. Exponential decay is modeled by functions of the form
where 0<b<1. b is called the decay factor.
Note: Aekt is another model for exponential decay if the instantaneous growth rate, k, is negative.
Powered by Desmos
A logarithmic function has the form f(x)=logax, for a>1. The domain of f is x>0, and the range is R:
Powered by Desmos