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.
We're going to graph some logarithmic functions. First of all, I want to recall some properties of logarithms, there are three of them I want to talk about. First, the log of a product, the log base b of x times y is the log base b of x plus the log base b of y.
The log of a quotient, log base b of x over y, equals the log base b of x minus the log base b of y. And the log of a power, log base b of x to the n equals n times log base b of x.
Here is a problem, I want to graph two functions f of x equals log base 2 of x over 8 and g of x equals log base 2 of x minus 3. Now before I do this, I want to examine these functions a little bit closer. I'll start with f of x log base 2 of x over 8. Now by the second property of logarithms, this can be written as the log base 2 of x minus the log base 2 of 8. And the log base 2 of 8 is 3, so the log base 2 of x over 8 is the same as log base 2 of x minus 3. These two functions are the same function, so I need only to graph one of them.
The first thing I'm going to do is I'm going to make a table of values for the log base 2 of x and then I'll do the subtracting of 3 later. Log base 2 of x, by the definition of logs is the same as x equals 2 to the y. And so the way I'm going to plot points is I'm going to pick values of y first and then I'll calculate the x values. I want to pick easy values like -1, 0, 1.
When y equals -1, x is 2 to the -1 which is 1/2, when y equals 0, x is 2 to the 0 which is 1, and when y equals 1, x equals 2 to the 1 which is 2. Let me plot those points really quickly. (1/2, -1), (1, 0) and (2, 1). I'm going to go ahead and graph y equals log base 2 of x. This is not the function I wanted to graph, but it will be really helpful to have a picture of it; y equals log base 2 of x. All I have to do is take this graph and shift it down three units to get the graph of f of x, y equals log base 2 of x minus 3. So let me choose a different color here.
So I'll take each point and shift it down three units, so this point goes down 1, 2, 3, this point goes down 1, 2, 3, and well it's all plotted. So let me draw a nice smooth curve here, this will be y equals log base 2 of x minus 3, that's it.
When I'm graphing log functions, especially when there are eddy functions, when there is a translation involved, I'll often draw the untranslated graph first and then I'll shift it down. When you have more than one graph in your picture, you might want to label them so that your teacher knows that you know which of these is which.