Voiceover:What I want to do in this video is graph up a classic exponential function and then graph a related

logarithmic function and see how the two are related visually. The two things I’m going to graph are y is equal to two to the x power and y is equal to the log base two of x. I encourage you to pause the video, make a table for each of them and try to graph them

on the same graph paper. See how they are related and if you see how they’re related, think about why they are related that way. Let’s first start with y

equals two to the x power. I’m going to make a little table here, different x values and the

corresponding y values. x and y, we can start with negative two, negative one, zero, one, two, three. In each case y is going to

be two raised to these power. Two to the negative two power is going to be 1/4. Two to the negative one power is 1/2. Two to the zero power is one. Two to the first power is two. Two to the second power is four. Two to the third power is eight. Let’s graph that. Two to the third power is eight. Two to the second power is four. Two to the first power is two. Two to the zeroth power is one. Two to the negative one power is 1/2. Two to the negative two power is 1/4. Even the two to the negative third power is going to be 1/8, so it’s going to look something like this. The graph is going to

look something like this right over here. It’s kind of your classic, sometimes this will be called

your exponential hockey stick because it kind of looks

like a hockey stick where it just kind of starts kind of slow and just oohh bam, shoots straight up. Notice as we go to the left as x becomes more and

more and more negative our value approaches zero

but never quite gets there. If we have two to the

negative one millionth power it’s going to be a

very, very small number, very, very close to zero but it’s not going to be quite zero. We’re going to have a

horizontal asymptote at y is equal to zero or the x-axis is a horizontal asymptote. Fair enough. Now let’s graph y is equal

to log base two of x. Before I graph that, let’s

just think about another way of representing it. This literally says, for any x, what power, what exponent

y if I raise two to that would give me x. This is an equivalent statement as saying two to the

y power is equal to x. If you notice, what we’ve done here between these two things you’re essentially just

switching the x’s and the y’s. Here’s two to the x power is equal to y. Here’s two to the y power is equal to x. Really this and this you’ve

swapped the x’s and the y’s. What we will see is

that we can essentially swap these two columns. x and y, so let me just do 1/4, 1/2, one, two, four, and eight. Here now we’re saying if x is 1/4, what power do we have to raise two to, to get to 1/4. We have to raise it to

the negative two power. Two to the negative one

power is equal to 1/2. Two to the zero power is equal to one. Two to the first power is equal to two. Two to the second power is equal to four. Two to the third power is equal to eight. Notice all we did, as we essentially swapped

these two columns, so let’s graph this. When x is equal to 1/4, y

is equal to 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 to look like this. Notice, I think you

might already be seeing a pattern right over here. These two graphs are essentially the reflections of each other. What would you have to reflect about to get these two? Well you’d have to reflect

about y is equal to x. If you swap the x’s and the y’s, another way to think about, if you swap the axis you

would 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 to think about it is we swapped the x’s and y’s. Just as this, as x becomes more and more and more and more negative you see y approaching zero. Here you see is y is becoming

more and more negative as x is approaching zero, or you could say as x approaches zero y becomes more and more and more negative. The whole point of this is just to give you an appreciation for the relationship between

an exponential function and a logarithmic function. They’re essentially

inverses of each other. You see that in the graphs, they’re reflections of each other about the line y is equal to x.

very useful stuff, we are just starting to go through graphs and logarithms, these videos are helping quite a lot! thanks a lot sal! 🙂

Khan you are my hero and my saviour, i learned so much things about EVERYTHING with your videos

i realy love your videos keep on going like you do 🙂

brilliant. thanks for this!

THANK YOU!!!! U are a life saver.

Which is the name of software you use?

You are not matthew

so I'm assuming this is why you have to tack on a ln(of the base) when taking the derivative of a exponential function raised to a function.

You need to quit, quit, you got to quit repeating, repeating yourself so much, stop repeating yourself so much. This makes the videos very hard to listen to. If I do not get something I would rather rewind than hear you repeat so much!

….lÓk.mń.

so helpful.. thank you!

So easy yeah 😁😁😁😍 thank u sir

Soooooo helpful. Thank you