Exponential and Logarithmic Functions
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I want to do some examples of graphing logarithmic functions, but before I begin, let's recall the definition of logarithmic functions. Y equals log base b of x means, x equals b to the y. B is the base of the logarithmic function.
Here is my problem; I'm going to graph y equals log base 3 of x and y equals log base a third of x, how are the two graphs related? Let me answer that second question first. I want to analyze this function because this 1/3 here scares me a little bit. So I'm going to use the definition of logs to rewrite this in exponential form. So I have y equals log base b of x, b in this case is 1/3, I'm going to get x equals 1/3 to the y.
I know we remember from properties of exponents that 1/3 is 3 to the -1 and then the power to a power rule, x equals 3 to the -y. But then I can use the definition of logs again because I have an exponential expression here. I can use the definition of log backwards. Right now I have x equals 3 to the -y, that's going to give me -y equals log base 3 of x, and of course that means y equals negative log base 3 of x. This is huge.
I wanted to find out how the two graphs were related first because, now that I've discovered that y equals log base 1/3 of x is the same as minus log base 3 of x, I know that I can just graph y equals log base 3 of x. The other graph is going to be a reflection across the x axis. So let me do that, let me start by graphing y equals log base 3 of x. And again whenever I graph logarithmic functions, I always switch to exponential form first. So this is the same as, x equals 3 to the y. I'll make a table and just choose easy values.
Now the easiest way to do this is to start with the y values, so I'm going to start with -1, 0 and 1, the x values will be x equals 3 to the -1, 1/3, x equals 3 to the 0, 1, and x equals 3 to the 1 which is 3. So I'll use these points to graph my function of y equals log base 3 of x. So (1/3,-1) is here, (1,0) is here and (3,1) is here and then I draw a nice smooth curve connecting these. Again y equals log base 1/3 of x is the same as y equals negative log base 3 of x, so all I have to do is take these points and flip them across the x axis and that will give me my graph of y equals log base 1/3 of x. So this point, flipped across the x axis goes here. This point stays put and this point goes down here. So here is my graph of y equals log base 1/3 of x. Let me label them.
A couple of things to remember when you're graphing the log functions, first the definition of log, I used it a couple of times. And second, make sure that you know your properties of logs and of exponents because you might be able to notice a connection between the graph you're graphing and an easier function. Here I was able to manipulate y equals log base 1/3 of x into a function I find easier to graph.