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The Graph of a Logarithmic Function

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Today we're going to talk about graphs of logarithmic functions. So we're going to start with the graph of an exponential function-- something we already know about-- and we're going to use that to draw conclusions about the graphs of a logarithmic function. So let's start by looking at a graph of an exponential function and we'll see how we can come up with the graph for a logarithmic function. So I've got this generic exponential equation of y is equal to b to the x, some basis b to the x, and a graph that it could correspond with.

So we know that logarithmic functions and exponential functions are inverse of each other. And we know that with inverse functions we can swap the x and the y variable or the input and the output. So this exponential equation y is equal to b to the x can be written as x is equal to b to the y. And now I can write this equation in logarithmic form. And that would become log base b of x is equal to y.

So now this is an equation that I can determine the graph for. And again, I'm going to do that by thinking about the relationship of inverse functions. And I know that with interest functions, they are symmetric above the line y equals x. And y equals x is a diagonal line that goes through every point on the graph where the x in the y-coordinates are equal to each other or the same. So that line looks approximately like this. Every point on this line, the x and the y-coordinates are the same.

And if the inverse functions, exponential and logarithmic, are symmetric on this line, above this line, that means that I could reflect my exponential graph over that line to determine my logarithmic graph. So doing that I know that this is going to have an x-intercept of 1. And then as I approach 0 on my x-axis, my y values are approaching infinity, negative infinity. And as I approach positive infinity on my x-axis, my y values are also approaching positive infinity. So my graph of logarithmic function will look something like this. And the equation is going to be y is equal to log base b of x.

So let's look at the domain and the range of a logarithmic function. We just saw that the graph of a logarithmic function is inverted from the graph of an exponential function. They're symmetric across the line y equals x. And we will also see a similar scenario with the domain and the range. So the domain and the range of a logarithmic function is opposite the domain and the range of an exponential function.

So let's start by looking at the domain of the logarithmic function. We see from the graph that it is restricted to positive values of x. So as we approach 0 on the x-axis, we see that our line is tending closer and closer to this line x is equal to 0, but it will never touch it. And it will never go past it into the negative side. So our values for x for a logarithmic function cannot be negative or equal to 0. They're restricted to positive values of x.

And if we're looking at our range, we see that as y approaches negative infinity, x is approaching 0. And as y is approaching positive infinity, x is also approaching positive infinity. So we can see that our range is all y values. So we can have y values that are negative. We can have a y value equal to 0. And we can have y values that are positive.

So finally, let's look at the y-intercept of our exponential graph and use that to draw conclusions about the x-intercept of our logarithmic graph. So for our exponential graph, I know that any exponential equation written in this form y is equal to b to the x has a y-intercept at the point 0, 1. And that's for a couple of different reasons. The first is that because we know that any base b to the 0 power is going to be equal to 1. And that's for any non-zero value of b. And we can see that that matches our equation, because if I have a value of 1 for y, then my x value is going to be 0.

We also then can convert this into logarithmic form and that would be log base b of 1 is equal to zero. And this is true for all positive values of b. And so because of these two facts, we can see that it makes sense that any exponential equation written in this form will have a y-intercept of 1.

We can use that to think about our x-intercept for a logarithmic equation. So any logarithmic equation written in this form, y is equal to log base b of x, is going to have an x-intercept at the point 1, 0. And this, again, is for the same reasons that b to the 0 equals 1 for any non-zero values of b. And because log base b of 1 is equal to 0 for all positive values of b.

So let's go over our key points from today. The graph of an exponential function reflected over the line y equals x is the graph of a logarithmic function. This is because exponential and logarithmic functions are inverse of each other. The domain of a logarithmic equation, y is equal to log base b of x, is all real x values greater than 0. And the range is all real y values. And the graph of a logarithmic equation, y equals log base b of x, has an x-intercept of 1, 0. And this is because 0 is equal to log of 1.

So I hope that these key points and examples helped you understand a little bit more about the graphs of logarithmic functions. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.