1.4 Exponential and logarithm graphs. Example 1. Recall that b = 2 a if and only if a = log 2 (b) That tells us that the functions f(x) = 2 x and g(x) = log 2 (x) are inverse functions. It also tells us that the graphs y = 2 x and y = log 2 (x) are reflections of each other in the line y=x. (a) This is moment to stop and reconstruct this chain and make sure you understand how it works. Use the diagram at the right to tie the reflection property and the (a, b) equivalence at the beginning together. Locate a point (a, b) on the graph y = 2 x and locate the corresponding point (b, a) on the graph y = log 2 (x). Are the two points reflections of one another in the line y = x? [Yes or no? Is this easy to see? Is it obvious?] (b) The same is true of course for other bases. The graphs y = 10 x and y = log 10 (x) are also inverse graphs and reflect into one another through the diagonal. On the same set of axes, draw the graphs y = 10 x and y = log 10 (x) so that we can compare these with the base-2 graphs drawn above. (c) Can you find a base k with the property that the graphs y = k x and y = log k (x) intersect one another? How many intersections can you get? One? two? Draw reasonable graphs of sample configurations. Solution. (a). It is clear that a point (a, b) is on the graph of a function f if and only if the point (b, a) is on the graph of its inverse function g. What we need to clearly see is that (a, b) and (b, a) reflect into one another through the line y = x. My favorite way to see that is to draw the coordinate box for each point. It s clear that each box reflects into the other (because horizontal line segments reflect into vertical line segments of the same length) and so the two points have to follow. exp&log graphs 10/6/2007 1
(b) The graphs y = 10 x and y = log 10 (x) are plotted at the right. The 10 x graph hugs the negative x-axis more closely than the 2 x graph but then as it as it crosses the y-axis it rises more steeply. In the same way, log 10 graph hugs the negative y-axis more closely than the log 2 graph and then as it as it crosses the x-axis it rises less steeply. (c) As we reduce the base below 2 (trying plots for base 1.9, 1.8, 1.7, etc.) we see the two curves getting closer together and we get an intersection for 1.4. Thus the graphs y = 1.4 x and y = log 1.4 (x) intersect twice, at x close to 2 and just above 4. Of course they intersect on the line y = x. As the base is reduced, there ought to be a point at which the curves k x and log k (x) first intersect and at that point they will have to be tangent to one another and to the line y = x. Through experimentation, we find the transitional base to be approximately k = 1.45. The configuration is graphed at the right. In fact, those who go on to study calculus, will be able to identify this critical base as e 1/e 1.445. exp&log graphs 10/6/2007 2
Example 2. Below, find two boxes, each containing the graph of a function. The graph on the left belongs to the function y = 2 x on the interval [0, 5]. The graph on the right is defined on the interval [5, 30] but it follows exactly the same curve in the sense that if one box was laid precisely over the other the curves would coincide. Find the equation of the graph in the right-hand box. Students find this problem hard, or at least perplexing, but it does not have to be if it is approached in a careful, organized manner. The graphs are the same but the axes are altered. This alteration is of two flavours: a stretch (or compression) and a shift. A good idea is to do one of these at a time, and the one to start with is the stretch. The width has gone from 5 to 25, a factor of 5, and the height has gone from 40 to 4, a factor of 1/10. So a good first step (to cut our teeth, so to speak) would be to find the equation of the graph which simply multiplies the axes by 5 and 1/10. That s the graph at the right below. The x-axis is expanded by a factor of 5. That requires a multiplication of x by some factor k. Should k be 5 or 1/5? The answer is that k should be 1/5. The best way to see this is to be a bit abstract and call the function on the left f(x). Then f, as an abstract function, is defined only on [0, 5], so when you write a formula for the graph on the right, do you want to have f(5x) or f(x/5)? The answer is clearly f(x/5) so that as x runs from 0 to 25, x/5 runs from 0 to 5 and we can legitimately apply f. If we have used 5x, we d be in trouble. Thus to expand the axis by a factor of 5, we have to multiply x by 1/5. The same argument works for the y-axis. Here we have a dilation of the axis by a factor of ¼ so we multi- x ply y by 4. The new equation is 4y = f which in this case is 5 4y = 2. 5 It remains to do the shift and this works in the same opposite way. We need to now add 5 to all the values on the x-axis and we do this by subtracting 5 from x. And we need to add 1 to all the values on the y- axis and we do this by subtracting 1 from y. We get the equation: Use a few specific values of x to check. x 5 = 5 4( y 1) 2. x exp&log graphs 10/6/2007 3
Example 3. Below I have sketched graphs of 13 members of the family y = r x belonging to the r values: r = 1/5, 1/3, 1/2, 1/1.6, 1/1.4, 1/1.2, 1, 1.2, 1.4, 1.6, 2, 3, 5 If you drew a vertical line at x=1, starting on the x-axis, you d meet these curves one by one in the above order. Note that all these r-values are positive. The function is not defined for r negative or zero. The purpose of this exercise is to allow the students to get better acquainted with this important family. In answering the questions below, they should be encouraged to give complete well-written answers and explanations. (a) Some of these curves are increasing and some are decreasing. What r-values belong to each set and why? (b) There is a significant symmetry in the above diagram. Describe this as precisely as you can and explain why it works. (c) Discuss the nature of the solution set of the equation r x = x For any fixed r > 0. That is, for any particular r, how many solutions, x, are there and where are they located? I m not asking you to find the solutions for any particular r, but to describe how they change as r changes. Note to the teacher on part (c). Some students might not have handled a request quite like this before and will wonder what s required. I expect the students to draw the graph y = x, and ask how many intersections it has with the r-curve for each r. A good answer will say that for any r < 1, there is a unique solution, positive, less than 1, and increasing with r. For r = 1, there is a unique solution x=1. For r > 1, there are two solutions if r is not too big. As r increases, the smaller solution increases and the larger solution decreases, until at some transitional value r*, they fuse together in a burst of flame and then die, so that for r > r*, there are no solutions. Some students will use a graphing calculator to show that r* is close to 1.45. In fact, as reported in Example 1 it turns out that r* = e 1/e. That s a good problem for the calculus course. exp&log graphs 10/6/2007 4
Problems 1. At the right is the graph y = (4/3) x. Using a construction on this graph make as good an estimate as you can of: (a) (4/3) 3 (b) log 4/3 (3) (c) A solution to the equation (4/3) x = log 4/3 (x). Work as carefully and neatly as you can. Use a sharp pencil or pen and a ruler. 2. At the right are the graphs y = 2 x and y = log 2 (x). (a) On the same set of axes, draw the graphs y =(1/2) x and y = log 1/2 (x). (b) Having done (a), you have four graphs on the same set of axes. Describe as simply as you can the geometric relationship between each pair of graphs. [With 4 graphs there are 6 pairs, so your answer has 6 parts.] 3. As in Example 2, the graph on the right belongs to the function y = 2 x on the interval [0, 5]. If the box on the right framing the graph is turned upside-down and placed on top of each of the three boxes below, the two curves will coincide. Use this to find equations of each of the three graphs. (a) (b) (c) exp&log graphs 10/6/2007 5
4. To get the graph at the right I rotated the graph y = log 2 (x) through 180. (a) Write down an equation for this graph. (b) Write down an equation for the inverse of the graph and sketch its graph. 5. Below is a sketched of the graphs of 13 members of the family y = r x belonging to the r values: r = 1/5, 1/3, 1/2, 1/1.6, 1/1.4, 1/1.2, 1, 1.2, 1.4, 1.6, 2, 3, 5 On a new set of axes, draw graphs of a number of members of the family y = r x x. Show clearly any asymptotes and find their equations. 6. Since 30 is halfway between 20 and 40, wouldn t you expect log30 to be halfway between log20 and log40? If not, should it be bigger or smaller? I want you to make this argument graphically, but instead of using the graph y = log(x), I want you to do it with the graph y=10 x, a copy of which is drawn at the right. Work carefully and neatly on the graph (use a ruler!) and write full explanations. exp&log graphs 10/6/2007 6