4. One important characteristic of exponential functions is that they grow at a faster rate than polynomial functions. You can see this illustrated here in this short page from Illustrated Mathematics(Links to an external site.). What that means is that no matter what polynomial function ()f(x)you write down, any exponential function ()h(x)will, past a certain value of x, "overtake" ()f(x). In other words, eventually ()h(x)will start to give larger output values than ()f(x), for the same input values.

a) Consider the polynomial function()=10f(x)=x10 versus ()=10h(x)=10x. Determine where the exponential function overtakes the polynomial function. Find that point of intersection graphically, using Desmos. Zoom in and note the point of intersection.

5. Using a graphing utility such as Desmos and the change-of-base property, work together to reach a consensus on the following.

First, graph ()=log3,()=log25,()=log100f(x)=log3x,f(x)=log25x,andf(x)=log100x in the same viewing window.

a) Which graph is on top in the interval (0, 1)? b) Which graph is on the top in interval (1, )? c) Generalize by writing a statement about which graph is on top, which is on bottom, and in which intervals, using ()=f(x)=logbx where >1b>1.

6. Does the graph of a logarithmic function have a horizontal asymptote? A vertical? Explain.

7. What type(s) of transformations(s), if any, affect the domain of a logarithmic function? The range? To explore this, graph the simple logarithmic function ()=5f(x)=log5xand its transformations