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# Logarithmic Functions - Problem 1

###### Norm Prokup

###### Norm Prokup

**Cornell University**

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.

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###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

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