1,000,000 ~ 106 ~ one million

100,000 ~ 105 ~ one hundred thousand

10,000 ~ 104~ ten thousand

1,000 ~ 103 ~ one housand

100 ~ 102 ~ one hundred

10 ~ la' ~ ten1 ~ 10° ~ one

0.1 ~ 10-' ~ one tenth0.01 ~ 10-2 ~ one hundredth

0.001 ~ 1O-3~ one thousandth

0.0001 ~ 1O-4~ one ten thousandth0.00001 ~ 10-5 ~ one hundred thousandth

0.000001 ~ 10-6~ one millionth

log 106 ~ 6

log 105 ~ 5log 104 ~ 4

log 1 03 ~ 3

log 102 ~ 2log 10' ~ 1

log 10° ~ alog 10" ~ -1log 10-2 ~-2

log 10-3 ~-3log 10.4 ~-4

log 10-5 ~-5log 10-6 ~-6

Figure 13.2. Common Logarithms of Powers ofTen.

A. Common LogarithmsCommon logarithms (also called logs or 10glO) areclosely related to scientific notation, The common logof a number is the power to which 10 must be raised togive that number, Figure 13,2,

100 = 102

The log of 100 is 2 because 10raised to the second power is 100

log lff = 2 .

1,000,000 = 106The log of 1,000,000 is 6 because 10raised to the sixth power is 1,000,000

log 106 = 6

0,001 = 10-3The log of 0.001 is -3 because 10raised to the -3 power is 0.001

log10-3 = -3

1 = 10°The log of 1 is 0 because 10

raised to the zero power is 1 (by definition)log 10° = 0

So, by definition, log (lOX) = X

To what power must 10 be raised to equal the number57 This is not obvious. We can reason that the number 5is between 1 and 10 so the log of 5 must be greater thanthe log of 1 (which is 0) and less than the log of 10(which is 1). The same is true for the numbers 2,3,4,6,7, 8, and 9; their logs are decimals between 0 and 1.There is no intuitive way to know the exact log of 5,although we know it is between 0 and 1. Rather, it isnecessary to look up the log in a table of logarithms, orto find it using a scientific calculator. The log of 5 isapproximately 0.699, which means that 10°·699= 5.Thus:

5 = lOXTake the log of both sides:

log 5 = XFrom a calculator we find that:

CHAPTER 13 BASIC MATH TECHNIQUES 223

X = 0.699which means that:

5 = 10°·699

The same logic can be applied to numbers between10 and 100. For example, what is the log of 487 The logof 10 is 1; the log of 100 is 2. Therefore, the log of thenumber 48 must fall between 1 and 2. It is, in fact,approximately 1.681.

log 48 = XFrom a calculator we find that:

X = 1.681which means that:

48 = 101.681

All numbers between 10 and 100 have a log between 1and 2. We can continue and apply the same logic to allpositive numbers. For example, the log of 4,987,000must fall between 6'and 7 because 4,987,000 is between1 million (106)and 10 million (107).

log 4,987,000 = XX = 6.698

which means that:106.698= 4,987,000

What about numbers between 0 and 17 Observe inFigure 13.2 that the log of 1 is 0 and that the logs ofnumbers between 0 and 1 are negative numbers. There-fore, all numbers between 0 and 1 have a negative log.For example:

The log of 0.130 is approximately -0.886which means that:10-0.886= 0.130

The log of 0.00891 is approximately -2_05which means that:10-2.05 = 0.00891

Note that 0 and numbers less than 0 do not have logs atall because there is no exponent we can use that willmake 10 to that number equal 0 or a negative. Formally,we say that logarithms of 0 or negative numbers areundefined.

B. AntilogarithmsAn antilogarithm (antilog) is the number correspondingto a given logarithm. For example:

The log of 100 is 2; therefore, 100 is the antilog of 2The log of 5 is approximately 0.699; therefore, 5 isapproximately the antilog of 0.699

What is the antilog of 37 Remember that logs areexponents; therefore, to find the antilog of a number, n,use n as an exponent on a base of 10. If n = 3, thenantilog 3 = 103 = 1000. 1000 is the antilog of 3.

224 MATH IN THE BIOTECHNOLOGY LABORATORY:AN OVERVIEW

What is the antilog of 2.5?Antilog 2.5 = 102.5

bil it is not obvious what 102.5 equals, we can reasona the antilog of 2.5 must be a number between 100

and 1000. (If this is not clear, think about the "2" in 2.5.)A scientific calculator is the simplest way to findantilogs. On many calculators the "antilog key" is thesecond function of the "log key." (Consult your calcula-tor directions to determine how to perform this func-tion.) Using my calculator I can find the antilog of 2.5 intwo ways:

2.52nd

log

The answer 316.227766 appears.Alternatively I can press:

10yX2.5=

The answer 316.227766 appears.Your calculator might work in a slightly different order;try it out.

EXAMPLE PROBLEM

What is the antilog of 3.S8?

ANSWER

We can reason that the answer is between 1000 and10,000 because. of the 3.

Antilog (3.58) = 103.58 = 3801.89

EXAMPLE PROBLEM

What is the antilog of -0.780?

ANSWER

The antilog of a negative number must be a valuebetween 0 and I.

Antilog ( - 0.780) = 10-0.780 = 0.166

c. Natural Logarithms

Common logarithms have a base of 10. There are alsologarithms whose base is ~ 2.7183, called e. When e isthe base, the log is called a natural log, abbreviated In.There are tables of natural logs and there are keys oncientific calculators to find the natural log. Note that

the terms log and In are not synonyms and should neverb used interchangeably. Natural logs are less com-monly used in biotechnology laboratories than logswith a base of 10.

D. An Application of Logarithms: pHThe proper function of aqueous biological y te -depends on the medium having the correct concentra-tion of hydrogen ions. pH is a convenient means [0

express the concentration of hydrogen ions in a solution.The concentration of H+ ions in aqueous solutions

normally varies between 1.0 M and 0.00000000000001 1\1(where M is a unit of concentration). The number0.00000000000001 can also be written as 1 X 10-14.Seren Serenson devised a scale of pH units in which thewide range of hydrogen ion concentrations can be writ-ten as a number between 0 and 14. If the hydrogen ionconcentration in a solution is 1 X 10-14 M, then usingSorenson's method we say its pH is 14. If the hydrogenion concentration of a solution is 0.0000001 M, or 1 X

10-7 M, then we say its pH is 7. If the hydrogen ion con-centration of a solution is 1 M, then we say its pH is O.Thus, to find the pH of a solution:

Step 1. Take the log of the hydrogen ion concentra-tion (expressed as molarity).

Step 2. Take the negative of the log.

For example:

What is the pH of a solution with aH+ concentration of O.000234 M?

Step 1. The log of 0.000234 is approximately - 3.63.

Step 2. The negative of - 3.63 is 3.63. So, 3.63 is thepH of this solution.

We can express the relationship between pH andhydrogen ion concentration more formally using thefollowing expression:

pH = -log [H+]

The symbol [ J means "concentration." In words, thisexpression means "pH is equal to the negative log of thehydrogen ion concentration."

If we know the pH and want to find the [H+] in asolution, then we apply antilogs. To do this:

Step 1. Place a negative sign in front of the pH value.Step 2. Find the antilog of the resulting number.

For example:

What is the concentration of hydrogen ionsin a solution whose pH is 5.60?

Step 1. Place a - sign in front of the pH: -5.60.

Step 2. The antilog of -5.60 is = 2.51 X 10-6. Theconcentration of hydrogen ions in this solu-tion, therefore, is 2.51 X 10-6 M.

The relationship between hydrogen ion concentra-tion and pH is expressed more formally using the fol-lowing expression:

[H+)'= antilog (-pH)

------

IPLE APPLICATION ANSWER (from p.222)

e CI dissociates to release 1.2 X 10-3 M of hydro-gen ions. The log of 1.2 X 10-3 is = -2.9.The negativeof this log value is 2.9. So the pH of the stomach after ameal is about 2.9.

MANIPULATION PRACTICEPROBLEMS: LOGARITHMS

L Answer the following without using a calculator:a. Is the log of 445 closer to 2 or 4?b. Is the log of 1876 closer to 3,9, or 10?

2. Give the common log of the following numbers.a. 100 b. 10,000 c. 1,000,000d. 0.0001 e. 0.001

3. The log of 567 must lie between 2.0 and 3.0. For eachof the following numbers, state the values the logmust lie between.a. 7 b. 65,9 c. 89.0 d. 0.45 e. 0.0078

4. Use a scientific calculator to find the log of the fol-lowing numbers.a. 1.50 X 104

d. 2.98 X 10-5b.345e. 1209

c. 0.0098f. 0.345

5. Use a scientific calculator to find the antilog of thefollowing numbers.a. 4.8990 b. 3.9900 c. -0.5600 d. -0.0089e. 9.8999 f. 1.0000 g. 8.9000

6. Use a scientific calculator to convert each of the fol-lowing [H+] to a pH value.a. [0.45 M] b. [0.045 M]c. [0.0045 M] d. [0.00000032 M]

7. Use a scientific calculator to determine the hydro-gen ion concentration of the solutions with the fol-lowing pH values.a. 4.56 b. 5.67 c. 7.00 d. 1.09 e. 10.1

III. UNITS OF MEASUREMENT

EXAMPLE APPLICATION:UNITS OF MEASUREMENT

The bacterial toxin botulinum is extremely toxic andacts by paralyzing muscles. Exposure to a few micro-grams of toxin causes death. The toxin. however. can beused to treat patients with a rare disorder. blepharo-spasm. These patients are blind because their eyes aresqueezed shut. Minute doses of the botulinum toxinallows patients to open their eyes. If a patient is admin-istered 100 pg of toxin. will the patient be able to see or

'II the patient die?

Answer on p. 226

CHAPTER 13 BASIC MATH TECHNIQUES 225

Scientists and technicians often measure things suhas the length of insects, the concentration of pollutanin an air sample, or the amount of air in a patient's lungMeasurement is basically a process of counting Forexample, consider an insect that is 1.55 em in length.Length is the property that is measured, 1.55 is the valueof the measurement, and centimeters (em) are the unitsof measurement. Another way to write this measure-ment is: 1.55 X 1 em. This latter statement tells us thatthe length of the insect was determined by comparisonwith a standard of 1 em in length and was found to be1.55 times as long as the standard. Measuring length istherefore comparable to counting how many times thestandard must be placed end-to-end to equal the lengthof the object. Another example is the characteristic oftime that can be measured by counting how many timesthe sun "rises" and "sets." The standard in this case is theapparent motion of the sun and the units are days.

All measurements require the selection of a stan-dard and then counting the number of times the stan-dard is contained in the material to be measured. 'Theterms standard and unit are related, but they can be dis-tinguished from one another. A unit of measure is a pre-cisely defined amount of a property, such as length ormass. A standard is a physical embodiment of a unit.Centimeters and millimeters are examples of units ofmeasure; a ruler is an example of a standard, Becauseunits are not physical entities they are unaffected byenvironmental conditions such as temperature orhumidity. In contrast, standards are affected by theenvironment. A strip of metal used to measure meterswill differ in length with temperature changes and willtherefore only be correct at a particular temperatureand in a particular environment.

A group of units together is a measurement system.The measurement system common in the United States,which includes miles, pounds, gallons, inches, and feet, iscalled The United States Customary System (USCS). Inmost laboratories and in much of the world, the metricsystem, with units including meters, grams, and liters, isused, see Figure 13.3 on p. 226.

The units in the U.S, system are not related to oneanother in a systematic fashion. For example, for length,12 in = 1 ft; 3 ft = 1 yd. There is no pattern to theserelationships. In contrast, in the metric system, there isonly one basic unit to measure length, the meter. Theunit of a meter is modified systematically by the addi-tion of prefixes. For example, the prefix centi means1/100, so a centimeter is a meter/l Gi). Kilo means 1000,so a kilometer is a meter X 1000. The same prefixes canbe used to modify other basic units, such as grams orliters. The prefixes that modify a basic unit always rep-resent an exponent of 10, see Table 13.1 on p, 226,

The most important basic metric units for biologistsare meters (m), for length;grams (g),for weight;and liters(L), for volume. Table 13.2 (p. 227) shows how prefixesare used to modify the basic units. Table 13.3 on p. 227