In this video we're going to have a look at exponential and logarithmic functions. So what they look like and how they relate to each other. So an example of an exponential function is y equals e to the x, the most common exponential function. And an exponential function just means we have some number here and e is a number, it's Euler's number, and it's e to the power of and x will be our power.
Now what this will look like is, well let's have a think. e is a number between 2 and 3, it's 2.7, and if we give x some values, we can plot what this curve will look like. And if x is, for example, 0, e to the 0 is 1. So when x is 0, y will be e to the 0, which is 1, which is why e to the x passes through this point here, where this is 1. And then if x gets smaller...
For example, if we go into the negative values for x, we're going to get e to the power of negative numbers. Now, if we know our Indice Laws, a positive number, like e to the power of any negative number, it's actually still a positive number. e to the power of negative 5 is just 1 on e to the 5. And this is still a positive number, it's just very small.
So, which is why our exponential... our function e to the x actually gets smaller and smaller and closer to zero. But it never touches this line here, because we can't actually get a value for x where our y will be zero, because there's no value here where this will end up being zero. Now, if x increases, well, e to the power of 5 is bigger than e to the power of 3, and e to the power of 10 is bigger than e to the power of 5. So as x gets bigger, our function gets bigger.
And that's when we see we have exponential growth. We see a function like this. And things that have exponential growth are maybe the number of users on a popular website because you find out about the website and you tell your three closest friends and then they tell their three closest friends and everyone's telling their friends and the growth will have exponential shape to it because more and more people are finding out every time.
Okay, so that's what a basic exponential... function looks like and the most basic logarithmic function, I'll draw this one in red, would be y equals ln x, the natural log and this is the same as log base e of x. So hopefully we have seen what a log is, a logarithm and we have a base and we have a term.
Now what this function looks like is, I will draw it for you, when x is equal to 1 ln of 1 has a value. This pretty much means e to the power of what is 1, e to the what is 1, and it's going to be 0. So when x is 1, y is 0, which is y passes through here. When x is 1, y is 0. And what we're actually going to have a shape here of ln x is a very similar shape, but slightly different. it will look like this.
And you may have, you may think, oh, they look like a reflection of each other. And you would be spot on if you said that, because e to the x and ln x are actually reflection in this diagonal line here, y equals x. And if you, if you are good with functions, you will know if a function is reflected in y equals x, they are actually the inverse of each other. They are the inverse. So.
You may already know that, but y equals e to the x and ln x, they are the inverse of each other. Now, with exponential and logarithmic functions in IB maths, they often just give you some function, whether it be an exponential or a logarithmic, and they might have some numbers in front here, or in front of the x, or plus here, and they're just basic function transformations. So, if you know, if you have, let's use an example, if you have y equals 2, e to the x plus 1 minus 5. These are just a bunch of transformations. The minus 5 means it's a vertical shift down 5 units.
The x plus 1 means a horizontal shift 1 unit to the left and the 2 at the front just means a vertical stretch. So maybe if you're if you still need some work on this watch the video on transformations of functions but this will just change this blue line slightly. And most of the time in IB, these types of questions will be with your calculator, so you can just simply sketch this and make sure you get your intercepts correctly. Okay, so exponential functions will typically look like this, unless there's a negative sign in front of the x, and then it's going to be exponential decay.
If there's a negative sign in front of the whole function, it will be a flipped version of it. But the basic shape we need to know is that it will increase. and a logarithmic function will have a similar shape, but the reflection in the y equals x line. Okay, good luck.