Voiceover:What I want to vì in this clip is graph up a classic exponential function and then graph a relatedlogarithmic function and see how the two are related visually. The two things I'm going to graph are y is equal to two lớn the x power & y is equal khổng lồ the log base two of x. I encourage you to pause the video, make a table for each of them và try to lớn graph themon the same graph paper. See how they are related và if you see how they're related, think about why they are related that way. Let's first start with yequals two to lớn the x power. I'm going lớn make a little table here, different x values and thecorresponding y values. X and y, we can start with negative two, negative one, zero, one, two, three. In each case y is going tobe two raised to these power. Two to lớn the negative two power nguồn is going to lớn be 1/4. Two to lớn the negative one power nguồn is 1/2. Two to the zero power is one. Two khổng lồ the first power is two. Two lớn the second power nguồn is four. Two to the third power nguồn is eight. Let's graph that. Two to lớn the third nguồn is eight. Two khổng lồ the second nguồn is four. Two lớn the first power is two. Two to lớn the zeroth power is one. Two to lớn the negative one power is 1/2. Two to lớn the negative two nguồn is 1/4. Even the two khổng lồ the negative third power nguồn is going to be 1/8, so it's going to look something like this. The graph is going tolook something like this right over here. It's kind of your classic, sometimes this will be calledyour exponential hockey stick because it kind of lookslike a hockey stick where it just kind of starts kind of slow và just oohh bam, shoots straight up. Notice as we go khổng lồ the left as x becomes more andmore & more negative our value approaches zerobut never quite gets there. If we have two to lớn thenegative one millionth nguồn it's going khổng lồ be avery, very small number, very, very close lớn zero but it's not going lớn be quite zero. We're going khổng lồ have ahorizontal asymptote at y is equal lớn zero or the x-axis is a horizontal asymptote. Fair enough. Now let's graph y is equalto log base two of x. Before I graph that, let'sjust think about another way of representing it. This literally says, for any x, what power, what exponenty if I raise two to lớn that would give me x. This is an equivalent statement as saying two to lớn they power is equal to lớn x. If you notice, what we've done here between these two things you're essentially justswitching the x's & the y's. Here's two to lớn the x nguồn is equal to lớn y. Here's two lớn the y power is equal to lớn x. Really this và this you'veswapped the x's and the y's. What we will see isthat we can essentially swap these two columns. X và y, so let me just vì chưng 1/4, 1/2, one, two, four, & eight. Here now we're saying if x is 1/4, what power vị we have lớn raise two to, to lớn get lớn 1/4. We have to raise it tothe negative two power. Two lớn the negative onepower is equal to 1/2. Two khổng lồ the zero nguồn is equal to lớn one. Two to the first power is equal to two. Two lớn the second power nguồn is equal to four. Two to lớn the third power nguồn is equal to lớn eight. Notice all we did, as we essentially swappedthese two columns, so let's graph this. When x is equal to lớn 1/4, yis equal lớn negative two. When x is 1/2, y is equal to negative one. When x is one, y is zero. When x is two, y is one. When x is four, y is two. When x is eight, y is three. It's going lớn look lượt thích this. Notice, I think youmight already be seeing a pattern right over here. These two graphs are essentially the reflections of each other. What would you have lớn reflect about lớn get these two? Well you'd have to reflectabout y is equal lớn x. If you swap the x's and the y's, another way to lớn think about, if you swap the axis youwould get the other graph. It's essentially what we're doing. Notice it's symmetric about that line and that's because these are essentially the inverse functions of each other. One way khổng lồ think about it is we swapped the x's and y's. Just as this, as x becomes more and more và more and more negative you see y approaching zero. Here you see is y is becomingmore and more negative as x is approaching zero, or you could say as x approaches zero y becomes more và more & more negative. The whole point of this is just lớn give you an appreciation for the relationship betweenan exponential function và a logarithmic function. They're essentiallyinverses of each other. You see that in the graphs, they're reflections of each other about the line y is equal lớn x.
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Graphing logarithmic functions (example 1)
Graphing logarithmic functions (example 1)
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