The Calculus Diaries - Jennifer Ouellette [103]
That original number is the base; to take an exponential, you multiply the base by itself x number of times. The number of times you multiply it by itself is the power, represented in superscript: for example, 10 multiplied by itself 5 times would be written as 105. When the base is 10, you can also think of the power as denoting the number of zeros to the right of the initial 1. So an exponential function would be something like 2x, or 5x, where the exponent is the variable. A power would be something like x 2, x 5, or x 3, where the base is the variable. It’s an important distinction.
Since taking a logarithm undoes the work of the exponential, in general, the logarithm is just the number of digits in that number. Just as with exponentials, if we’re dealing with a perfect power of 10, for example, the logarithm is the number of zeros to the right of the initial 1: log(10) = 1, log(100) = 2, log(1,000) = 3, and so on. Or, to put it as generally as possible, log(10x) = x. The only catch is that you can’t take the logarithm of a negative number: no such animal exists. The logarithm inverts the exponential, but you can’t get a negative number with exponentials.
THE PLOT THICKENS
Back at the start of my foray into calculus, my physicist spouse, Sean, would leave simple problems on our home whiteboard for me to solve, like little mathy love notes. (Yes, we have a whiteboard at home. Doesn’t everyone?) The first set of problems focused on learning how to plot the points generated by specific functions onto a Cartesian grid, then connecting the dots to see the shape of the resulting curve (or “face” of the function). I quickly figured out this was much easier to do in a handy program called Grapher: You just plug in different values for the variable(s) in a given function, hit Return, and the correct curve miraculously appears. (You can do the same thing in Excel.)
It’s fun to play with Grapher, but frankly, I found it just as instructive to slowly plot out a few functions by hand. Many of us have difficulty grasping the notion of just what a function is: The textbook definitions, while technically correct, usually convey little actual meaning to nonmathy sorts like me. Literally taking a given function apart, point by point, and slowly rebuilding it again can help bridge that gap in communication.
Let’s plot the function f(x) = ax2 onto a Cartesian grid with the familiar x and y axes. Remember that f(x) is just another way of writing y for calculus purposes; so we’re working with y = ax2. The process is simple, if tedious. Assuming that a = 1, all we are doing is plugging in different values for x to get the corresponding value for y and plotting the point where they intersect onto the grid. I found it helpful to write down those initial values into columns first.
We already know this will be a parabola. I chose whole numbers, both positive and negative, for simplicity’s sake, but you can plug in any value for x along the real number line: positive, negative, fractions, and so on. (If you don’t include negative values, you only get half the parabolic curve.) Remember that the function technically comprises all possible values for x in that equation taken together—i.e., an infinite number of values. That would be tedious to plot indeed. But you can plug in enough values along the number line, plot out the corresponding points on the grid, and at some point you accumulate enough points that a definite curvy pattern emerges when you connect the dots.
I’ve described curves as representing the “faces” of functions, but those faces can have multiple expressions. Someone who is happy, sad, or angry will have the same basic features, but their faces can look quite different depending on the emotions they are experiencing. The same is true of functions. For instance, the constant