After watching this video lesson, you will be able to recognize exponential and logarithmic functions by looking at the end behavior of the graphs. You will also learn how the graphs change.
What happens when you put a mama bunny together with a daddy bunny? They start to multiply, literally. Rabbits are known for how fast they make babies, and then how quickly those babies make even more babies. In math, we call this exponential growth because we can describe their growth with an exponential function.
What are these functions? They are the functions where our variable is in the exponent. That's easy to remember if you look at the first word and tell yourself that this first word is telling you where the variable is. So, we will have functions such as y = 2^x, y = 4^x, and y = 10^x. What do these look like when we graph them? Let's see:
Graph of exponential functions
The red line is the y = 2^x graph, the blue line is the y = 4^x graph, and the green line is the y = 10^x graph. Notice how the green line actually increases faster than the blue and red lines. Why do you think this is the case? This is because the base of our exponential function is bigger. We have a 10 instead of a 2 or a 4.
If these functions showed the population growth of two different groups of rabbits, we would say that the group represented by y = 10^x, the green line, has a faster growth rate. Since these functions are representing population growth, the base of our exponential function then represents the growth factor, or how fast our population grows.
Exponential End Behavior
What we are doing here is actually analyzing the end behavior, how our graph behaves for really large and really small values, of our graph. For exponential functions, we see that the end behavior tends to infinity really fast. The larger the growth factor, which is the base of the exponential function, the quicker we get to infinity. We also see that for very small values of our input, our variable, the graph is close to 0. For population growth, we don't worry about these values.
Now, let's look at logarithmic functions and how they are different from exponential functions. They are actually related to each other. Logarithmic functions are the functions where the variable is the argument of the log function. We will see functions such as y = log base 2 (x), y = log base 4 (x), and y = log (x). Remember that for logarithmic functions, if we don't see a base number, then it is automatically 10.
I said earlier that these functions are related to our exponential functions. They are in that logarithmic functions are inverses of exponential functions. Graphed, the logarithmic version will be the mirror image of our exponential function across the line y = x. Do you want to see? Okay, here goes:
Graph of logarithmic functions
In this graph, the red line is the function y = log base 2 (x), the blue line is the y = log base 4 (x) function, and the green line is the y = log (x) function. We have graphed the inverses of our exponential functions for our rabbit populations. As you can see, if we fold our graph paper diagonally through the origin, on the line y = x, then our logarithmic functions are the mirror images of our exponential functions.
Logarithmic End Behavior
Because the logarithmic functions are flipped exponential functions, their end behavior is a bit different. We see that we are limited to positive values for our input. As our input gets close to 0, our function drops to negative infinity. As our input gets larger and larger, the logarithmic function grows too, but slowly.
It doesn't grow as fast as the exponential, which is to be expected, since we are looking at the flipped version. We also see that the larger the base of our logarithm, the slower the growth is as well. It seems like our end behavior here is the opposite of our end behavior for our exponential functions.
Let's review what we've learned now. We learned that exponential functions are the functions where our variable is in the exponent, and logarithmic functions are the functions where our variable is the argument of the log function. Logarithmic functions are inverses of exponential functions. Examples of exponential functions are y = 2^x and y = 4^x. Examples of logarithmic functions are y = log base 2 (x) and y = log base 4 (x).
The end behavior of a graph is how our function behaves for really large and really small input values. For exponential functions, we see that our end behavior goes to infinity as our input values get larger. The larger the base of our exponential function, the faster the growth. For logarithmic functions, our function grows slowly as our input values get larger. The larger the base of our logarithmic function, the slower the growth.
Following this lesson, you'll have the ability to:
- Define exponential functions and logarithmic functions
- Explain the relationship between these types of functions
- Describe the end behavior of exponential and logarithmic functions