Graphing Logarithmic Functions - Video Tutorials & Practice Problems

Graphs of Logarithmic Functions

Hey everyone earlier, when graphing exponential functions, we found and plotted a bunch of different ordered pairs. And we found that our graph had similarities to both polynomial and rational functions. As we approached an Asymptote. Now we're faced with graphing another type of function, a logarithmic function. And you may be dreading having to learn how to graph yet another different type of function. But I have great news for you here because graphing a log function is actually super similar to graphing an exponential function because a log is the in of an exponential. So here I'm going to walk you through graphing a log function using exclusively things we already know about graphing exponential functions. So let's go ahead and get started. Now looking at the log function we have here, we have a log base two of X. So let's start by just plugging into values of X and getting some ordered pairs. So starting with X equals one. If I plug that into my function, I get log base two of one. And I know anytime I take the log of one, no matter what the base is, I'm always going to get zero. So I have my first ordered pair at 10, which I can go ahead and plot on my graph. Then if I plug in X equals two to my function, I get log base two of two, which since my base is the exact same as what I'm taking the log of. I know that my answer here is going to be one and I get my second ordered pair at 21. Now let's pause here and take a closer look at these two points. I have a 10 and 21. Now looking over to my exponential function here I have F of X is to two to the power of X which is the inverse function of this log function that we are trying to graph here. Now I looking at my exponential function, I have these two points 01 and 12 and looking back to the points that we just found, I noticed that these are the same exact points with X and Y flipped. Now this is because these are inverse functions. So this is actually going to work for every single X and Y value of our exponential function. If we simply with them, we'll have all of the new ordered pairs needed to graph our log function. So let's go ahead and flip our X and Y values and get these points plotted on our graph. So looking at our first point here 1/4 and negative two, having flipped those points there is my new point. And then for one half, I know F of X will be negative one. Now flipping these final two points here, I'll end up with 42 and 83. And we can go ahead and plot all of these on our graph. So 42, 83 and then for our small fractions, one half, negative one and 1/4 negative two. Now connecting all of these points, I have a complete graph of my log function. And I notice here on this side that we're getting really close to that Y axis but not quite touching it, which tells us that we're of course, dealing with an Asymptote that I can go ahead and plot using a dash line right along that Y axis to form my vertical Asymptote. Now that we have complete picture of the graph of our log function, let's compare it to our original exponential function. Now looking at the graph of my exponential function here, if I were to take my entire graph and fold it along this diagonal line and then stamp my exponential function onto the other side. I would end up with a graph identical to the log function that we just graphed. Because this is actually exactly what we did. We simply reflected the graph of our exponential function over this diagonal line Y equals X in order to get the graph of our log function. Now this will work for any log function in its corresponding inverse exponential function because they're inverses, they're always going to be reflections of each other along the line of Y equals X. Now, if you want a way to remember the shapes of these graphs, remember when you're dealing with an exponential function, you can think of a lowercase E. And if we just extend the tail of that E out, it looks really similar to the shape of the graph of our exponential function. And for a logarithmic, if we look at our word logarithmic, and we take that lowercase R and extend that R out, we get a shape really similar to the graph of our logarithmic function. So remember for an exponential function graph, the shape looks really similar to extending a lowercase E. Whereas for a logarithmic function, we extend that lowercase R now that we have a good idea of what's going on with the shapes of these graphs. And what they look like, let's also consider their domain and range. Now when graphing our log, we are able to flip the X and Y values of the inverse exponential function and we can actually do the same exact thing to our domain and range. So the domain of our exponential function corresponds to the range of our inverse log function. So it's going to be all real numbers for any log function of any base. Then also the range of our exponential function corresponds to the domain of our log function and it's going to depend on our Asymptote and in this case, go from zero to infinity. Now, speaking of asymptotes, we also can take a look at our Asymptote here. And we had a horizontal Asymptote at Y equals zero for our exponential function, which then corresponds to a vertical Asymptote. Now at X equals zero for our logarithmic function. Now let's consider one final thing here and consider when we have to graph logs of different bases. Now, everything that we've looked at so far has been kind of the opposite of working with an exponential function. But this thing is going to be the exact same. And the direction of our graph is going to depend on the value of our base B just as it did for exponential functions. And it's going to depend on it in the exact same way. So for values of B that are greater than one, like two or three or 10 or whatever our graph is going to be increasing. And for values of B between zero and one, like say one half our graph is instead going to be decreasing. Now that we know everything that we need to about graphing log functions. Let's get some more practice.

Graphing Logarithmic Functions

Hey, everyone in this problem, we're asked to grab two different functions F of X is equal to three to the power of X and G of X is equal to a log base three of X. Now, we can see that these functions are inverses of each other because the log is the inverse function of an exponential. And we're going to go ahead and graph both of them together. So let's go ahead and start with F of X is equal to three to the power of and find and plot some points here. So starting with X equals zero, we can plug that into our function and get three to the power of zero, which is just one. Now, if I plug one into my function that's three to the power of one, which is just three and then three to the power of two is going to give me nine. Now let's go ahead and plot these points on our graph. So I'm gonna start with 01 and then I have 1329 is completely off of my graph. Actually, we know that it's going to increase really rapidly and get really steep there. Now, for my negative values of X, we're gonna see the same numbers, but instead of fractions as we had with positive one and two. So three to the power of negative one is simply 1/3 and then three to the power of negative two is 1/9. And we can see that we're getting smaller and smaller as we go into those negative numbers. So let's go ahead and plot those points at negative 11 3rd and then 219 really close to my X axis. So we can go ahead and connect all of these points that we have on our graph here. I know that I'm gonna get really steep on this side. And then on my left side here, I'm gonna get really close to my X axis but not quite cross it because we have an Asymptote there. So I can go ahead and plot my Asymptote as well. I know that it's at that X axis right at Y equals zero. Now that we have our graph for three to the power of X, let's move on to plotting G of X where we have log base three of X. Now remember that whenever we're dealing with inverse functions, we can simply swap our X and our Y values. So these values that I had for X over here are simply going to become my Y values when working with my new function. That is the inverse and then my, my Y values over here are going to become the X values of my inverse function. So let's go ahead and swap all of those points so that we can get them plotted on our graph. So starting with this point, I'm gonna go ahead and switch that up. So now it is 1/9 negative two just reversing my xmiy. Then for a negative 11 3rd that becomes one third, a negative one 01 flipped is 10. Then I have 13 which I flip to get three comma one. And then my last point flipping that two and that nine, I get nine comma two. Now let's go ahead and plot all of these points for our function at G of X to get our final function up here. So let's go ahead and start with this A 10 and work our way down this way first. So 10 and then I have three comma one and then two comma nine goes off of my graph yet again. And then going the opposite direction into my fractions one third and negative one is gonna be right about here and then 1/9 and negative two, we're gonna be getting really close to our Y axis this time. So we can go ahead and connect all of these points to form our graph. And then we're getting really close to that Y axis which tells us that we're dealing with a vertical Asymptote up on this side, right, on our Y axis at X equals zero. So now that we have plotted both of our functions, we can go ahead and determine our domain and our range. Now, when dealing with our domain of our first function F of X is equal to three to the power of X, I can see that this is going to be any value of X, it can be all real numbers. Now, we know that our domain of our three to the power of X is going to match up with the range of our function of log base three of X. So I know that my range here is also going to be all real numbers which I can verify by taking a look at my function here. Now, for my range of my function three to the power of X, I need to go ahead and take a closer look and identify where my Asymptote is so that I can get an accurate range here. Now, when I consider my range, I know that we're not going to go past that Asymptote here. So my range is simply gonna go from my Asymptote at that zero point all the way to infinity. So my range here is simply from zero to infinity. Now, we know that our range is going to flip and become the domain of our other function of our inverse function. So that tells me that the domain of log base three of X is going to be from zero to infinity, which we can again verify by taking a look at our graph, we see that we can't cross this Asymptote. So we start at that zero point and go to everything positive. Now that we've graphed these inverse functions. Let's get some more practice. Let me know if you have any questions.

Transformations of Logarithmic Graphs

Hey, everyone, we just learned how to graph basic log functions using the fact that they're the inverse of an exponential function. But what if we have to graph a more complicated log function like this G of X we have here, we might be worried that this is where it's going to start to get more challenging and switch up from exponential functions. But you don't have to worry about that at all because we're going to do the same thing we've done a million times before and not do any calculations, but just some transformations. So let's go ahead and jump right in now, our function G of X here looks really similar to log base two of X but just has a couple extra things going on here. This one and this four, both of which represent transformations. Now just as a quick recap of the most common transformations, remember that if you have a negative on the outside of your function that represents a reflection over the X axis, whereas a negative on the inside of your function represents a reflection over the Y axis, then of course, we have H which represents a horizontal shift and K which represents a vertical shift. Now, just as we did for exponential functions, when working with log function, we are always going to want to graph our parent function first because the base of your log function isn't always going to be the same. So let's go ahead and do that for this function G of X that we have here and start with step zero and plot that parent function first. So here we want to go ahead and identify our parent function I mentioned that in G of X, it looks really similar to log base two of X because that actually is our parent function here. So our parent function F of X is equal to log base two of X. We want to go ahead and graph 3.3 points here at one over B where B is our base. So in this case, one half negative one, then our second point will be at 10 which will be the same no matter what the base of your log function is. And then our last point will be at B that base one. So B here is two. So that last point is at 21. Now these are going to be the easiest points for us to plot and they're going to serve as sort of test points as we graph our new function. So plotting those on my graph, I have one half negative one, then 10 and then 21. Now I can go ahead and connect those points to get my graph here. And then I can also go ahead and plot my vertical Asymptote at X equals zero in order to get my entire parent function all ready to go and ready to be transformed. Now that we have that parent function, we can go ahead and start plotting our actual function G of X here starting with step one and shifting that vertical Asymptote to X equals H. Now here looking at my function, I have that H is this one because remember X minus H so here H is just positive one. So I can go ahead and shift my vertical Asymptote right over to that one using of course a dash line because it is an Asymptote. OK. Now that we have finished step one, we can move on to step two and identify whether or not there is a reflection happening. Now, remember a reflection happens if there is a negative on the inside or outside of our function. And here looking at my function G of X, I don't have a negative that got added. So I don't have to worry about that reflection and I can go ahead and move on to the second part of step two and shift my test points by H and K. Now we already identified H as being positive one and looking over at my function here, this negative four tacked on the end here represents K. So I'm gonna shift my test points of my function here by one point to the right and four points down. So let's go ahead and do that. Starting with this first point, I'm gonna go one to the right and 1234 down ending up right here. Then my next 0.1 to the right and 1234 down and my final 0.1 over to the right and 1234 down. So I have my new points here shifted by H and K. And then I can go ahead and move on to the very last step in plotting my function which is going to be to sketch my curve approaching the Asymptote. So here I can go ahead and connect all of my new points and then approach that Asymptote on the bottom there. Now that we have our complete picture of our graph, we can go ahead and identify our domain and our range as well. Now remember our range is always going to be the for our log functions no matter what that base is or what other shifts of transformations are happening. It's always going to be all real numbers. So there's my range, all real numbers. And then of course, my domain actually depends on my Asymptote and which direction I'm approaching my Asymptote from. So looking at my graph here, I'm approaching my Asymptote from that right side. And when that happens, that means that my domain is going to go from my Asymptote at H to infinity. Now, here H is one. So my domain is going to be from one to infinity. Now, if my, my graph was approaching my Asymptote from the left side instead, over here, my domain would then go from negative infinity until I reach that Asymptote at H. Now that we know how to graph log functions using transformations, let's get some more practice.

Graph the given function.

$g\left(x\right)=\log_2\left(x-1\right)-4$

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