Supplemental Lecture (97/04/24 update) by Stephen T. Abedon (abedon.1@osu.edu)

  1. Chapter title: Exponents and Logarithms
    1. A list of vocabulary words is found toward the end of this document
    2. Exponent notation and logarithms are used extensively in biology. Make sure you know how to use exponents and logarithms. Note, however, that I'm giving this review simply because when I say, for example, 10-7 or 10-7 * 10-7 = 10-14 or talk about exponential growth, I want you to have a clue as to what I am talking about. More long range, think of knowing how to use exponents and logarithms as nearly as important as knowing how to use the metric system.
  2. Exponents
    1. For The term xy, where both x and y are numbers, y is called an exponent. In English, this expression translates to "the number x raised to the number y." More plainly, the expression xy is equal to the number 1 multiplied by the number x a total of y times. So, if x = 2 and y = 3, xy = 1 * x * x * x = x * x * x.
    2. Note that y need not be an integer. Yes, that makes things a little confusing. However, solving expressions such as (6.1) 0.34 is what scientific calculators exist for (answer = 1.9 if rounded to a single significant figure). To the point, it is far more important that you can reach a point of being able to represent your answer as an exponent than that you can solve complicated expressions off the top of your head.
  3. Powers of 10
    1. One expects of a scientist an understanding of what 10 raised to a given integer equals, e.g., 10-1 = 0.1, 103 = 1000, etc. More completely:
      1. 10-10 = 0.0000000001 (note that there are 9 zeros to the right of the decimal point)
      2. 10-9 = 0.000000001
      3. 10-8 = 0.00000001
      4. 10-7 = 0.0000001
      5. 10-6 = 0.000001
      6. 10-5 = 0.00001
      7. 10-4 = 0.0001
      8. 10-3 = 0.001
      9. 10-2 = 0.01
      10. 10-1 = 0.1
      11. 100 = 1
      12. 101 = 10
      13. 102 = 100
      14. 103 = 1000
      15. 104 = 10000
      16. 105 = 100000
      17. 106 = 1000000
      18. 107 = 10000000
      19. 108 = 100000000
      20. 109 = 1000000000
      21. 1010 = 10000000000 (note that there are 10 zeros to the left of the decimal point)
  4. Scientific notation
    1. The most common use of exponents in science (hence the first half of the name) and the most important reason to get used to using exponents. In scientific notation numbers are represented as the product of a real number with only a single digit to the left of the decimal and the number 10 raised to an appropriate value. Thus: 602300000000000000000000 = 6.023 x 100000000000000000000000 There are 23 zeros to the left of the decimal in the right-most value. Thus: 100000000000000000000000 = 1023 and 602300000000000000000000 = 6.023 x 1023
    2. Similarly, all of the following expressions convey the same value and amount of information (though, strictly, only 6.023 x 1023 is in scientific notation):
      1. 0.0006023 x 1027
      2. 0.006023 x 1026
      3. 0.06023 x 1025
      4. 0.6023 x 1024
      5. 6.023 x 1023
      6. 60.23 x 1022
      7. 602.3 x 1021
      8. 6023. x 1020
      9. 60230 x 1019
      10. 602300 x 1018
      11. 6023000 x 1017
  5. Laws of exponents
    1. Once a number is reduced to scientific notation, it is very easy to manipulate arithmetically. Thus:
      1. 6.023 x 1023 x 10 = 6.023 x 1024
      2. 6.023 x 1023 / 10 = 6.023 x 1022
      3. 6.023 x 1023 / 6.023 = 1.0 x 1023
      4. 6.023 x 1023 x 6.023 x 1023 =
      5. = (6.023 x 1023)2
      6. = (6.023)2 x (1023)2
      7. = 36.28 x 10(23*2)
      8. = 36.28 x 1046
      9. = 3.628 x 1047
    2. More generally:
      1. x0 = 1, where a 1 0
      2. kxy = xyk; -kxy = -xyk
      3. (-x)y does not always equal -(x)y
      4. xy + mn = mn + xy
      5. xy * mn = mn * xy
      6. xy + xz = xz + xy
      7. xy * xz = x(y + z)
      8. xy / xz = x(y - z)
      9. 1 / xy = x0 / xy = x(0 - y) = x(-y)
      10. xy / mn = xy * m-n
      11. (xy)z = x(y * z)
      12. (xy)z = xzyz
  6. Negative exponents
    1. As seen above, a negative exponent refers to the inverse of a number raised to a positive integer. Thus, a negative exponent should immediately suggest that the number in question has a value of less than one. For example, 2-3 = 1 / (2 * 2 * 2) = 1/8 = 0.125. Working with powers of 10:
      1. 10-5 = 1/105 = 0.00001
      2. 10-4 = 1/104 = 0.0001
      3. 10-3 = 1/103 = 0.001
      4. 10-2 = 1/102 = 0.01
      5. 10-1 = 1/101 = 0.1
      6. 100 = 1/100 = 1
      7. 101 = 1/10-1 = 10
      8. 102 = 1/10-2 = 100
      9. 103 = 1/10-3 = 1000
      10. 104 = 1/10-4 = 10000
      11. 105 = 1/10-5 = 100000
  7. Fractional exponents
    1. You are already familiar with fractional exponents. They are not a big deal. For example, square root of 4 = 40.5 = 2. Also, 80.33 = 2. In fact (25)0.2 = 2(5 * 0.2) = 2. Get it?

  8. Exponential functions

    1. The expression y = zx solved for y in terms of x will not graph as a straight line on normal graph paper. In other words, the progression z0, z1, z2, z3, . . ., zn does not increase in value linearly (and, in fact, not at all for all values of z less than or equal to 1). Instead, the above function progresses exponentially. That is, each subsequent value is equal to the previous value multiplied by a constant, z. Thus, z0 = 1, z1 = z0 * z = z, z2 = z1 * z = z * z, etc. For z 1 this gives a graph that resembles that presented below.
  9. Illustration, exponential growth

  10. Log transformation
    1. Note what happens with the same graph when graphed on what is known as log-linear graph paper (or, rather, its computer generated equivalent, below).
  11. Illustration, exponential growth [log-linearly graphed]

    1. Note that now the graph is linear. What happened? Essentially the same as what happens in the graph (below) if we logarithmically (log10) transform the y axis.
  12. Illustration, exponential growth (log transformation)

    1. In other words, logz(zx) = x (and above is a graph of log10; y = log10(10x)).
  13. Natural logarithm [ln]
    1. Natural logarithms are simply logz where z = e = 2.7182818 . . . Don't worry about this except to be aware that if you haven't already run across e and natural logarithms, you will.
  14. Laws of logarithms
    1. The following are general laws of logarithms:
      1. logz(XY) = logz(X) + logz(Y)
      2. logz(X/Y) = logz(X) - logz(Y)
      3. logz(Xy) = ylogz(X)
  15. Vocabulary
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  16. Practice questions
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  17. Practice question answers
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  18. References
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