Log problem

Question

On a graph, graph the function f(x)=a^x and g(x)=log sub a x when a=2, a=.5 and a=1. how many points of intersection does each value of a produce ? 

Answer 1

On my TI-84 Plus calculator, it doesn't recognize any log functions that are not the common log, or base 10 log function. So you have to know the change of base formula, which is log(X)/log(a). 

 

a=2:

Plot Y1=2^X

Plot Y2=log(X)/log(2)

There are no points of intersection.

 

a=0.5

Plot Y1=0.5^X
Plot Y2=log(X)/log(2)
There is one point of intersection, at about x=0.641.

 

a=1

Plot Y1=1^X

This graph produces the line Y=1.

Y = Log sub1 of X can not be graphed electronically because it is not a function. In the change of base formula, log(1)=0, and you can not divide by 0. Remember a function has only 1 y-value for each x-value. On paper, you can work out a few points to plot: (1,0), (1,1), (1,2) etc. You get the line x=1. There is one point of intersection, at (1,1).