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**The Use of and Power of Using Semi-Log Graph Paper**

This kind of graph paper allows you to graph exponential data without having to translate your data into logarithms. The paper does it for you! At stationery and university bookstores, you can buy semilog graph paper with anywhere from 1 to 5 or 7 cycles. "Cycles" will be explained in the next paragraph. For those of you who are likely to use many different kinds of graph paper, including semi-log and log-log graph papers, you might be interested in purchasing a book, from which you can photocopy the specific type you need at the moment.

Shown here is what is known as 3-cycle semilog graph paper. You will notice that the vertical axis is very peculiar as the numbers only go up from 1 to 9 and start all over with 1 again, over and over. This is because the distances indicate logarithmic distances. And you will remember that there is no log of zero! (If you want to print out a full-page copy of the graph, click on the graph to the right.)

As shown in the next figure, you MIGHT consider the line across the bottom as equal to one, and the next horizontal line labelled as 1 should be ten, and the next 1 is 100 and the top line is 1000. The lines' basic numeric value may not be changed. The only thing allowed is the placement of the decimal point - and they always differ by only one decimal place per cycle. Thus you might have 0.001, 0.01, 0.1 and 1.0, or you might have 10^{5}, 10^{6}, 10^{7} and 10^{8}. So you can see that semilog graph paper can plot both very small numbers as well as astronomical ones.

Let's see how this works and what the supposed power of the semilog graph is all about. In brief it is terrific for plotting anything that is exponential - such as compound interest.

Suppose that you make a series of numbers such that each one is double the one before it (black). (Or you might make each one 3 times the value of the previous one (red); or, for fun, 1.5-times the previous one (blue).) Starting with the series that is of the powers of two (the black line), we find 2^{0} (=1) and mark it at the lower left corner; then move to 2^{1} (=2), and plot that on the second major vertical line; then move to 2^{2} (=4) and plot that on the next major vertical line, and so on with 2^{3} (=8), 2^{4} (=16), and so on. And - AMAZINGLY - we get a straight line. Scientists love straight lines! If something gives a straight line on semilog graph paper, we call it an exponential function.

You will see that the 3^{x} and the 1.5^{x} also give straight lines. Wonderful! Any type of exponential function can be plotted to give a straight line!

For any values that increase exponentially with time, one can use these graphs to easily determine doubling times (the blue figures associated with the red line). All you have to do is take any point on the red line, and then go move upwards to a point that is double that value and see how much horizontal distance was gained. That is called the doubling time, or t_{1/2}. Of course, for convenience, you wouldn't choose a value like 22.86 and have to find twice that. Rather you would choose 1.00 and easily find 2.00, as was done in this diagram. Also shown are some successive doublings to 4 and then to 8. You will notice that the horizontal distances remain the same for each of the sections.

In reality, you are more likely to get data that plot like the green line. This is called a growth curve in microbiology. For awhile, in the "log phase" the microbes are growing exponentially (see the straight portion of the line?), but then they begin to slow down because on any number of reasons or combination of reasons - buildup of waste products, exhaustion of food, etc. Soon growth ceases and the line runs horizontally in "stationary phase." Quiz: what is the doubling time for this microbe during exponential growth? (Answer: it doubles about once per hour.) Another name for doubling time is "generation time" or just plain "g".

**SURVIVAL and Radioactive DECAY CURVES**

The difference between growth curves as shown above and survival curves is from which corners the line or lines radiate. Above they have come from or near the bottom left and gone upwards. Survival curves come from the upper left and go downwards. That corner is defined as "1" or 100%. (And remember that the log of 1 = zero!) If you are using three cycle semilog paper, that means the upper cycle goes from 100% to 10%, the middle cycle is from 10% to 1%, and the lower cycle from 1% to 0.1%. For more on this click button!

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