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Graph of Logarithmic Functions - Concept

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!

In science classes we will often find ourselves graphing logarithmic functions to describe situations such as motion or speed over time. When trying to identify these situations as those seen in graphing logarithmic functions, it is important to be able to recognize these graphs. It is also important to recognize graphs of exponential functions and their importance as the logarithmic inverse.

Finding the graph of a logarithmic function. So for this example I'm not actually going to give you the graph straight up. What we're going to actually do is talk about how we get it, okay? So think about any time we are dealing with a equation in logarithmic form. What we generally do is put it into exponential form. So I'm going to do that first. So this is just going to give us x=2 to the y.
Now think back to actually how we got to this formula in the first place. And that was by taking the inverse of an exponential. So the inverse is then just when we switch our x and y's. y=2 to the x. Okay.
We know what that graph looks like, okay and for this I have a little prop for you. We have the inverse sorry the graph of y=2 to the x looks something like this. Okay? It's not precise but pretty rough. How we found the graph of an inverse was by flipping something over the line y=x. So what that did is it flipped everything that was above the line down, everything that was below it up. And what you end up with is a graph that looks like this, okay. So by using properties of inverses, we can actually go from the graph that we have already derived to the graph of the log function, okay? So what this graph does, the exponential graph went to the point 0, 1. When we flip the x and y's we are now going to go to the point 1,0. Our exponential graph had a horizontal asymptote at 0 that gets flipped to be a vertical.
So now I have a vertical asymptote at what is that x or y or is it x=0, okay. So vertical vertical line right here and then our domain, our x values we weren't able to ever get to 0 before so we're still not going to be able to do that for our domain, 0 to infinity. And our range used to be our domain for our inverse function, which was everything and so this is just going to be obvious. So this is a rough sketch of a log graph.
Key thing to note is this is only if our base is greater than 1, okay? This exponential graph that we looked at the first one was if our base was greater than one so that translates to this this base having the same restrictions. Okay? So by using our properties of inverses we're able to find the rough sketch of our log graph with our base bigger than 1.

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