Graphing logarithmic functions (example 1)

- [Instructor] We are told the graph of y is equal to log base two of x is shown below, and they say graph y is equal to two log base two of negative x minus three. So pause this video and have a go at it. The way to think about it is that this second equation that we wanna graph is really based on this first equation through a series of transformations. So I encourage you to take some graph paper out and sketch how those transformations would affect our original graph to get to where we need to go. All right, now let's do this together. So what we already have graphed, I'll just write it in purple, is y is equal to log base two of x. Now the difference between what I just wrote in purple and where we wanna go is in the first case we don't multiply anything times our log base two of x, while in our end goal we multiply by two. In our first situation, we just have log base two of x while in here we have log base two of negative x minus three. And in fact we could even view that as it's the negative of x plus three. So what we could do is try to keep changing this equation and that's going to transform its graph until we get to our goal. So maybe the first thing we might want to do is let's replace our x with a negative x. So let's try to graph y is equal to log base two of negative x. In other videos we've talked about what transformation would go on there, but we can intuit through it as well. Now whatever value y would have taken on at a given x-value, so for example when x equals four log base two of four is two, now that will happen at negative four. So log base two of the negative of negative four, well that's still log base two of four, so that's still going to be two. And if you were to put in let's say a, whatever was happening at one before, log base two of one is zero, but now that's going to happen at negative one 'cause you take the negative of negative one, you're gonna get a one over here, so log base two of one is zero. And so similarly when you had at x equals eight you got to three, now that's going to happen at x equals negative eight we are going to be at three. And so the graph is going to look something like what I am graphing right over here. All right, fair enough. Now the next thing we might wanna do is hey let's replace this x with an x plus three, 'cause that'll get us at least, in terms of what we're taking the log of, pretty close to our original equation. So now let's think about y is equal to log base two of, and actually I should put parentheses in that previous one just so it's clear, so log base two of not just the negative of x, but we're going to replace x with x plus three. Now what happens if you replace x with an x plus three? Or you could even view x plus three as the same thing as x minus negative three. Well we've seen in multiple examples that when you replace x with an x plus three that will shift your entire graph three to the left. So this shifts, shifts three to the left. If it was an x minus three in here, you would should three to the right. So how do we shift three to the left? Well when the point where we used to hit zero are now going to happen three to the left of that. So we used to hit it at x equals negative one, now it's going to happen at x equals negative four. The point at which y is equal to two, instead of happening at x equals negative four, is now going to happen three to the left of that which is x equals negative seven, so it's going to be right over there. And the point at which the graph goes down to infinity, that was happening as x approaches zero, now that's going to happen as x approaches three to the left of that, as x approaches negative three, so I could draw a little dotted line right over here to show that as x approaches that our graph is going to approach zero. So our graph's gonna look something, something like this, like this, this is all hand-drawn so it's not perfectly drawn but we're awfully close. Now to get from where we are to our goal, we just have to multiply the right hand side by two. So now let's graph y, not two, let's graph y is equal to two log base two of negative of x plus three, which is the exact same goal as we had before, I've just factored out the negative to help with our transformations. So all that means is whatever y value we were taking on at a given x you're now going to take on twice that y-value. So where you were at zero, you're still going to be zero. But where you were two, you are now going to be equal to four, and so the graph is going to look something, something like what I am drawing right now. And we're done, that's our sketch of the graph of all of this business. And once again, if you're doing it on Khan Academy, there would be a choice that looks like this and you would hopefully pick that one.