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.
Hi, I want to talk about a really important logarithm, the common logarithm.
The common log is just a log base 10 of x, although it's so much that it's actually just written log of x. So whenever you see log of x, that usually means the log base 10.
In this example I want to graph a related logarithm, the log base root 10 of x. Before I do that I'm going to do a little of Algebra because this root 10 scares me a little bit. So I'm going to change the base on this logarithm; y equals log base root 10 of x. Just by the definition of logarithm, is the same as root 10 to the y equals x.
Now root 10 is the same as 10 to the 1/2 and so by the power to a power property of exponents, this is the same as 10 to the y over 2 equals x. Now 10 to the y over 2 equals x, I can use the definitions of logarithms again to rewrite this in logarithmic form and I get y over 2 equals the log base 10 of x, or just the log of x. And finally multiplying by 2, y equals 2 log x. So the graph of y equals log base root 10 of x is the same as twice the log of x and I'll use this fact when I'm graphing it.
I'm going to make a table of values for the log base 10 function first. Y equals log base 10 of x means 10 to the y equals x, so let me plot a few points here. I'm going to use easy numbers -1, 0 and 1, numbers that are easy to exponentiate with the base 10. 10 to the -1 is point 1, 10 to the 0 is 1, 10 to the 1 is 10. So that gives me three points of the graph of y equals log x. Let me plot those.
The final graph that I'm looking for is going to be a vertical stretch of this graph, so I'll graph that second. So let me plot these points. Now here this is going to be x equals 1, so (1,0) goes here. The (.1, -1) is going to be way in tied against the y axis right like there. And then the point 10, 1 all the way up to 10; 2, 3, 4, 5, 6, 7, 8, 9, 10, (10, 1) is right there. This is a really shallow logarithm, kind of hard to draw.
Now the function I wanted to graph is y equals 2 log x, so let me change colors. On each point, the y value will be twice the y value of this graph. So this point has a y value of -1, twice that is -2. Twice the y value of 0 is 0, so this point stays put and twice the y value of 1 is 2. So my transformed graph is going to look like this. This top graph is our final graph; y equals log base root 10 of x.
So remember, if you are working with a logarithm base root 10 or some other base that you don't like, you can always change the base by using the definition of logarithm, and use properties of logs or transformations to graph your function.