 This video is going to talk about logarithmic graphs. So there's only two types of logarithms that we can put in our calculator, common logs which are base 10 and natural logs which are base E. So lots of times we have problems that are some other base and how we're supposed to graph those. If we take the log base being converted to an exponential and we can try to graph that one and then see what happens. So we have this log base 2 of x and we say that that's 2 and we say that that's 2 half across equals sign to the y is equal to x. But remember that in inverse functions then the y's and the x's change places. So this is really 2 to the x is equal to y and that's what we're going to graph. Now we're going to do this by hand. So let's remind ourselves that we had 2 to the x is equal to y. And I think I actually want to add at least one more data point in here now that I think about it. So let's let x be negative 1. 2 to the negative 1 would be 1 over 2 to the first or 1 half. 2 to the 0, anything to the 0 is 1. 2 to the 1 is just going to be 2. And then 2 to the 2 is 2 squared or 4. So let's graph those points. Negative 1, 1 half is similar about there. 0, 1, 1, 2, and 2, 4. So we have this exponential curve that's going to look something like this. Now it says switch the roll of x and y to graph the logarithm. Well if we take those same values that we had in that table and make the y's become x's and the x's become y's. My x's were negative 1, 0, 1, and 2, so now I'm going to let those be my y values. So negative 1 over here and the y value is 0, 1, and 2. And then for my x values, they are the y values. So negative 1 went with 1 half, so now my x is 1 half. 0 is with 1, so now x is 1, then we have 2, and then 2 went with 4. That's a 2. So 1 half, 1 would be this point about right there. 1, 0 would be on the x-axis. 2, 1, and then 4, 2, and we get a graph that looks something like this one. Now remember when we were talking about inverses, we said that they were mere images across the y equal x. And you can see that, again, we have these inverse functions going on here. And if we paid attention, this point right here is the 0.24, and the point like it on the graph right across from it, across the y equal x is the 0.42. So we have f of f graphed here, and we want to estimate when the graph is f of 3. Remember this 3 in here is an x, so we're finding y, and when x is 3 it looks like y is 1. And f of x equal 2, now remember this one, since it's not in the parentheses, x is, that's the y, and we're finding x, which is exactly what the problem said anyway. So that should be a clue not to look at the x-axis. So y equal 2, so we come up to 2 on the y-axis until we hit our graph, and it looks like it hits it somewhere around here, and this, we'll make it a nice number and call it 9. And it asks, does this graph have any asymptotes? Well, this graph is getting real close to this y-axis, and in fact, because of how thicker our line is, it looks like it might be crossing the y-axis, or at least becoming the y-axis, but it isn't actually. So our y-axis is actually an asymptote here, and that's a vertical line, so it would be x equal 0. Now finally, we have this f of x graph again, same graph, but we want to find the domain and range. The domain, remember we talked about the fact that this was an asymptote, so it was getting close to 0, but not really crossing it, that y-axis. So we go from 0, almost 0, that's why it's a parentheses, to infinity. And the range, this graph is going down forever. In fact, let me change colors there so you don't think I'm talking about the asymptote. This graph is going to go down forever, and then it's going to go up forever, its range would be all reals. Now it asks us, why do our answers make sense, based on the graph of the exponential? Well, if you think about the exponential, it would look something like this graph. Okay, this is just a rough sketch, it's not a perfect sketch, but it looks something like this. Remember that it's above the x-axis and doesn't cross it. So the range value, the y's above the x-axis, then it was close to 0 to infinity. And our domain, it went left and it went right forever, so it was all reals. And we know that what we just found for a domain being 0 to infinity makes sense because exponential range was 0 to infinity. And we know the range value in this function, this logarithmic graph, being negative infinity to infinity makes sense because the exponential domain was all reals.