Wed. May 30, 2007. Prepared by Mary Parker Based on Seidman’s Chapter 8. page 1 of 3

Logarithms.

“A logarithm IS an exponent.”

So, when we talk about logarithms, we must be clear what base we’re using.

Common logarithms:

Look at Seidman, p. 110. Read the top half of the first column. What base are we using here? Answer: Base 10

After that, consider one hundred thousand. Write it in the normal form, write it as a power of ten, and then find the logarithm of it. Discuss it with the person sitting next to you.

Now, let’s consider the log of 37,000,000. What two powers of ten is 37,000,000 between? 10,000,000 and 100,000,000. So that’s 10^7 and 10^8. So the log of it is between 7 and 8.

Now, use a calculator to find log(37,000,000). Does it agree with the size you expected? How would you check it? (Ans. Take 10 to that power. Does that equal 37,000,000? It should.)

Consider the log of 0.00837. What two powers of ten is 0.00837 between? Answer 0.0100 and 0.001, which is between 10^(-2) and 10^(-3). So the log of 0.00837 is between -2 and -3.

Use a calculator to find it. Does it agree with what you expected? Check it.

Other bases: Are there other logarithms? Yes, we can use any positive number as a base. Obviously, for our system of writing numbers, 10 is a very convenient base. When the idea of logarithms was first developed, it was with 10 as a base so that we could do convenient things with our numbers.

One other base is used quite a lot in science, but we won’t use it in this book. See the last part of the second column of p. 110. This is “natural logarithms.” When you are doing using this idea in applications that involve using calculus, then these natural logarithms make many things more convenient. (Draw a picture of the exponential functions, and mention the slope of the function at x = 0. If the base is e, then that slope is 1, and 1 is a very convenient value for that slope.)

What about going the other direction from logs, that is, antilogs? Well, from a mathematician’s point of view, if I want the antilog(1.678) I would do that as 10^1.678 = 47.64 (using a calculator). Or you can use the “antilog key.” But what I Wed. May 30, 2007. Prepared by Mary Parker Based on Seidman’s Chapter 8. page 2 of 3

just told you is exactly what the antilog key is doing. What does your calculator show as the antilog key? Try it and try using this exponent. Do you get the same answer?

Why do we use common logs? Answer: In modern times, a major reason is to be able to think of things that are very different in size all at the same time. It’s kind of like order of magnitude, but not limited to whole number orders of magnitude.

Two major applications: 1. pH, which is a way of talking about the acidity/alkalinity of a subsatance. 2. Richter scale, which is a way of talking about the amount of seismic energy released by an earthquake.

About the Richter scale, I have some relatives in the San Francisco area and so we sometimes talk about earthquakes. They have experienced quite a few earthquakes of level 3.5 intensity or about 4.5 intensity. They don’t really think of those as being very different. But to go from 4.5 to 5.5 is a really big change. (Search the Internet for the Richter scale and you can find a verbal description of what the various levels mean in terms of what shakes and what types of things are broken. I found a nice example in Wikipedia.)

The point is that the 4.5 earthquake is ten times as powerful as the 3.5 earthquake, but the 5.5 earthquake is a hundred times as powerful as the 3.5 earthquake.

They have to have a scale like that if they’re going to be able to quantify both the huge destructive force of major earthquakes and the small force of mild earthquakes. pH scale:

In biology and chemistry, we have the pH scale. Read p. 111, the first ten lines or so of the first column. Notice that there’s a big difference between the numbers 1.0 M and 0.00000000000001 M. And it’s a real pain to write all those zeros. So, Mr. Sorenson noticed that it was a lot more convenient to think of them in scientific notation and then to just look at the exponent on the ten. And then, since these exponents were negative, and it’s tedious to deal with negative numbers, he wanted to drop the negatives. So, his idea was to come up with a new measurement: pH is –log [H+]

Explain to each other what each part of that notation means. What does the H mean? What does the + after the H mean? What do the brackets mean? How do we take a log? And the minus in front winds up telling us to just dump the negative sign that taking the log gives us.

Example: If a substance has [H+] = 0.00389 then find its pH. Check with your neighbor. Did you get the same thing? Wed. May 30, 2007. Prepared by Mary Parker Based on Seidman’s Chapter 8. page 3 of 3

Example: If a substance has pH = 8.2, find its [H+]. Check with your neighbor. Did you get the same thing?

You may wonder why I chose to start with the Richter scale as an example of logs, since it isn’t really relevant to biotechnology. I did that because I think it’s easier to think of earthquake A being ten times as destructive as earthquake B or earthquake C being a hundred times as destructive as earthquake D. I have more trouble thinking of the difference in hydrogen ion concentrations as easily. Perhaps you can think of those more easily.

I’m curious – let’s ask those of you here who think a lot about biology and chemistry – do you get to the point where you think of those differences in hydrogen ion concentration in terms of 0.00001 M being one hundred times less concentrated than 0.001 M?