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.
In order to solve equations that contain exponentials, we need logarithmic functions. Logarithmic functions are the inverses of exponential functions. The properties of logarithms are used frequently to help us simplify exponential functions. Logarithmic functions have a unique set of characteristics and asymptotic behavior, and their graphs can be easily recognized if we know what to look for.
Today we're going to talk about logarithmic functions. Exponential functions which we've learned about previously are all one to one functions and that means that they're all invertible. They all have inverse functions and the way we get those inverses is just to switch x and y. x equals b to the y is the inverse of y equals b to the x. This is going to become our logarithmic function, so here's our definition of logarithmic functions. y equals log base b of x means x equals b to the y. This notation here, this is just the name of the function that gives me the y value to which I have to raise b to get x definition of log.
Now to see a wide variety of logarithmic functions graft at once I'm going to need to use geometer sketch pad, so let's take a look at that. Here in geometer sketch pad we can see the exponential function f of x equals 2 to the x and its inverse. The log base 2 of x, right now b is 2, b is the base of both functions I can change that let me increase it. So when b is 3 you get a much steeper exponential graph but a much less deep logarithmic graph. Let me see where that continues if we increase b to 4 you can also have values of b between 0 and 1, so if I switch to say b equals 0.5 and now I have a decreasing exponential graph, decreasing logarithmic graph again making the b value smaller closer to 0 will make a steeper exponential graph and a less steep logarithmic graph. Again here's why we don't like b to equal 1, the exponential function becomes horizontal the about logarithmic functions they always pass through the point 1, 0 they always have that x intercept.
If you plug in their base in this case 2.49 you get 1 and they always just as the exponential functions have as horizontal asymptote the x axis logarithmic functions have as a vertical asymptote the y axis. What are the domains and range of logarithmic functions, well you probably remember when we talked about inverse functions that, the domain, that when you invert a function the domain and range switch. So the range of the exponential function, the positive numbers becomes the domain of logarithmic functions, and the range, the domain of the exponential functions becomes the range of logarithmic functions.
Okay let's review what we've learned. The domain of a logarithmic function is the set of positive numbers that means you can't plug 0 into a logarithm, you can't plug negative numbers in. They're only defined for positive numbers, the range all real numbers you can get any kind of number out of a logarithmic function. We also notice the graph of a logarithmic functions always pass through the point 1, 0 at its x intercept and that they always have the vertical asymptote x equals 0 I'm sorry. Okay let's graph an example of a logarithmic function. Let's do an easy one y equals log base 2 of x. Now the first thing I usually do when I'm dealing with logarithms is I use the definition to re-write them in exponential form. This definition, so b is 2 I'm going to re-write log base 2 of x, y equals log base 2 of x in this form. x equals 2 to the y. It's easier to plot points that way, so next I make a table, x and y the easiest way to make a table of values for a logarithmic function is to start with the y values and use this definition here. So I'm going to pick values nice exponential values like negative 1, 0, 1 and to get the x value I calculate 2 to the y. 2 to the negative 1 is one half, 2 to the 0 is 1 and 2 to the 1 is 2. So I have 3 points that I can plot and graph my logarithm. So again they always pass through 1, 0, 2, 1 is this point, 1 half negative 1 and just draw a smooth curve connecting these, this is y equals the log base 2 of x.
I almost always use the definition of log when I'm graphing a logarithmic function, it's much easier to calculate exponents than to calculate logarithms. So make sure that you know the definition of logarithms, it's really important. And also make sure you remember like key characteristics of the logarithmic graph. The vertical asymptote x equals 0, the x intercept the domain just the positive numbers and the ranges all real numbers.