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Graph of Logarithmic Functions - Problem 1

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Finding the graph of a logarithmic function where our base is between 0 and 1. So for this I’m going to actually show you how using laws of inverses we can find the graph. Logarithmic equations in general what we've been doing this putting them in their exponential form. So same idea over here this becomes x is equal to 1 half to the y.

Remember we got this from taking the inverse of an exponential equation to begin with. To find the inverse we just switched our x and our y’s so we started with y is equal to 1/2 to the x. We know what this graph looks like this graphs looks something like this little guy right here who is decreasing down and to find the inverse, all we do is we flip something over the line y equals x.

So what that does is it takes this everything above gets flipped down everything below gets flipped up and this would be the graph of the inverse of this, namely the log function. So drawing that over here.

Before our graph went through the point 0,1 we are now going to go through the point 1,0 and we end up with something like this. Before we had a horizontal asymptote that gets turned into a vertical asymptote and the graph is going to be decreasing.

Domain, the x values, before we had from 0 up as our range, range turns into domain whenever you take inverses this is going to be 0 to infinity. Our range for our function used to be the domain. Our domain was everything, so now our range is everything.

So using properties of inverses to find the rough graph of a logarithmic function with base between 0 and 1.

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