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Number Exponential Form Common Logarithm
1000 10 3 3
10 10 1 1
1 10 0 0
0.1 10 −1 −1
0.001 10 −3 −3

To find the common logarithm of most numbers, you will need to use the LOG button on a calculator.

Rounding and significant digits

In reporting numerical data obtained via measurements, we use only as many significant figures as the accuracy of the measurement warrants. For example, suppose a microbiologist using an automated cell counter determines that there are 525,341 bacterial cells in a one-liter sample of river water. However, she records the concentration as 525,000 cells per liter and uses this rounded number to estimate the number of cells that would likely be found in 10 liters of river water. In this instance, the last three digits of the measured quantity are not considered significant. They are rounded to account for variations in the number of cells that would likely occur if more samples were measured.

The importance of significant figures lies in their application to fundamental computation. In addition and subtraction, the sum or difference should contain as many digits to the right of the decimal as that in the least certain (indicated by underscoring in the following example) of the numbers used in the computation.

Suppose a microbiologist wishes to calculate the total mass of two samples of agar.

4.38 3 _ g 3.002 1 _ ______ g 7.38 5 _ g

The least certain of the two masses has three decimal places, so the sum must have three decimal places.

In multiplication and division, the product or quotient should contain no more digits than than in the factor containing the least number of significant figures. Suppose the microbiologist would like to calculate how much of a reagent would be present in 6.6 mL if the concentration is 0.638 g/mL.

0.63 8 _ g mL × 6. 6 _ mL = 4.1 g

Again, the answer has only one decimal place because this is the accuracy of the least accurate number in the calculation.

When rounding numbers, increase the retained digit by 1 if it is followed by a number larger than 5 (“round up”). Do not change the retained digit if the digits that follow are less than 5 (“round down”). If the retained digit is followed by 5, round up if the retained digit is odd, or round down if it is even (after rounding, the retained digit will thus always be even).

Generation time

It is possible to write an equation to calculate the cell numbers at any time if the number of starting cells and doubling time are known, as long as the cells are dividing at a constant rate. We define N 0 as the starting number of bacteria, the number at time t = 0. N i is the number of bacteria at time t = i , an arbitrary time in the future. Finally we will set j equal to the number of generations, or the number of times the cell population doubles during the time interval. Then we have,

N i = N 0 × 2 j

This equation is an expression of growth by binary fission.

In our example, N 0 = 4, the number of generations, j , is equal to 3 after 90 minutes because the generation time is 30 minutes. The number of cells can be estimated from the following equation:

N i = N 0 × 2 j N 90 = 4 × 2 3 N 90 = 4 × 8 = 32

The number of cells after 90 minutes is 32.

Most probable number

The table in [link] contains values used to calculate the most probable number example given in How Microbes Grow .

A table is titled Most Probable Number Table. For each row, it states the number of tubes giving a positive reaction for a 5-tube set for 10 mL, 1 mL and 0.1 mL tubes, followed by the MPN per 100 mL, and the 95% confidence limits for low and high. For row 1, the reactions are 10 mL = 0, 1 mL = 0, 0.1 mL = 0; the MPN is <2, and the low and high confidence limits are <1 and 7. For row 2, the reactions are 10 mL = 0, 1 mL = 1, 0.1 mL = 0; the MPN is 2, and the low and high confidence limits are <1 and 7. For row 3, the reactions are 10 mL = 0, 1 mL = 2, 0.1 mL = 0; the MPN is 4, and the low and high confidence limits are <1 and 11. For row 4, the reactions are 10 mL = 1, 1 mL = 0, 0.1 mL = 0; the MPN is 2, and the low and high confidence limits are <1 and 7. For row 5, the reactions are 10 mL = 1, 1 mL = 0, 0.1 mL = 1; the MPN is 4, and the low and high confidence limits are <1 and 11. For row 6, the reactions are 10 mL = 1, 1 mL = 1, 0.1 mL = 0; the MPN is 4, and the low and high confidence limits are <1 and 11. For row 7, the reactions are 10 mL = 1, 1 mL = 1, 0.1 mL = 1; the MPN is 6, and the low and high confidence limits are <1 and 15. For row 8, the reactions are 10 mL = 2, 1 mL = 0, 0.1 mL = 0; the MPN is 5, and the low and high confidence limits are <1 and 13. For row 9, the reactions are 10 mL = 2, 1 mL = 0, 0.1 mL = 1; the MPN is 7, and the low and high confidence limits are 1 and 17. For row 10, the reactions are 10 mL = 2, 1 mL = 1, 0.1 mL = 0; the MPN is 7, and the low and high confidence limits are 1 and 17. For row 11, the reactions are 10 mL = 2, 1 mL = 1, 0.1 mL = 1; the MPN is 9, and the low and high confidence limits are 2 and 21. For row 12, the reactions are 10 mL = 2, 1 mL = 2, 0.1 mL = 0; the MPN is 9, and the low and high confidence limits are 2 and 21. For row 13, the reactions are 10 mL = 2, 1 mL = 3, 0.1 mL = 0; the MPN is 12, and the low and high confidence limits are 3 and 28. For row 14, the reactions are 10 mL = 3, 1 mL = 0, 0.1 mL = 0; the MPN is 8, and the low and high confidence limits are 1 and 19. For row 15, the reactions are 10 mL = 3, 1 mL = 0, 0.1 mL = 1; the MPN is 11, and the low and high confidence limits are 2 and 25. For row 16, the reactions are 10 mL = 3, 1 mL = 1, 0.1 mL = 0; the MPN is 11, and the low and high confidence limits are 2 and 25. For row 17, the reactions are 10 mL = 3, 1 mL = 1, 0.1 mL = 1; the MPN is 14, and the low and high confidence limits are 4 and 34. For row 18, the reactions are 10 mL = 3, 1 mL = 2, 0.1 mL = 0; the MPN is 14, and the low and high confidence limits are 4 and 34. For row 19, the reactions are 10 mL = 3, 1 mL = 2, 0.1 mL = 1; the MPN is 17, and the low and high confidence limits are 5 and 46. For row 20, the reactions are 10 mL = 3, 1 mL = 3, 0.1 mL = 0; the MPN is 17, and the low and high confidence limits are 5 and 46. For row 21, the reactions are 10 mL = 4, 1 mL = 0, 0.1 mL = 0; the MPN is 13, and the low and high confidence limits are 3 and 31. For row 22, the reactions are 10 mL = 4, 1 mL = 0, 0.1 mL = 1; the MPN is 17, and the low and high confidence limits are 5 and 46. For row 23, the reactions are 10 mL = 4, 1 mL = 1, 0.1 mL = 0; the MPN is 17, and the low and high confidence limits are 5 and 46. For row 24, the reactions are 10 mL = 4, 1 mL = 1, 0.1 mL = 1; the MPN is 21, and the low and high confidence limits are 7 and 63. For row 25, the reactions are 10 mL = 4, 1 mL = 1, 0.1 mL = 2; the MPN is 26, and the low and high confidence limits are 9 and 78. For row 26, the reactions are 10 mL = 4, 1 mL = 2, 0.1 mL = 0; the MPN is 22, and the low and high confidence limits are 7 and 67. For row 27, the reactions are 10 mL = 4, 1 mL = 2, 0.1 mL = 1; the MPN is 26, and the low and high confidence limits are 9 and 80. For row 28, the reactions are 10 mL = 4, 1 mL = 3, 0.1 mL = 0; the MPN is 27, and the low and high confidence limits are 9 and 80. For row 29, the reactions are 10 mL = 4, 1 mL = 3, 0.1 mL = 1; the MPN is 33, and the low and high confidence limits are 11 and 93. For row 30, the reactions are 10 mL = 4, 1 mL = 4, 0.1 mL = 0; the MPN is 34, and the low and high confidence limits are 12 and 93. For row 31, the reactions are 10 mL = 5, 1 mL = 0, 0.1 mL = 0; the MPN is 23, and the low and high confidence limits are 7 and 70. For row 32, the reactions are 10 mL = 5, 1 mL = 0, 0.1 mL = 1; the MPN is 31, and the low and high confidence limits are 11 and 89. For row 33, the reactions are 10 mL = 5, 1 mL = 0, 0.1 mL = 2; the MPN is 43, and the low and high confidence limits are 15 and 110. For row 34, the reactions are 10 mL = 5, 1 mL = 1, 0.1 mL = 0; the MPN is 33, and the low and high confidence limits are 11 and 93. For row 35, the reactions are 10 mL = 5, 1 mL = 1, 0.1 mL = 1; the MPN is 46, and the low and high confidence limits are 16 and 120. For row 36, the reactions are 10 mL = 5, 1 mL = 1, 0.1 mL = 2; the MPN is 63, and the low and high confidence limits are 21 and 150. For row 37, the reactions are 10 mL = 5, 1 mL = 2, 0.1 mL = 0; the MPN is 49, and the low and high confidence limits are 17 and 130. For row 38, the reactions are 10 mL = 5, 1 mL = 2, 0.1 mL = 1; the MPN is 70, and the low and high confidence limits are 23 and 170. For row 39, the reactions are 10 mL = 5, 1 mL = 2, 0.1 mL = 2; the MPN is 94, and the low and high confidence limits are 28 and 220. For row 40, the reactions are 10 mL = 5, 1 mL = 3, 0.1 mL = 0; the MPN is 79, and the low and high confidence limits are 25 and 190. For row 41, the reactions are 10 mL = 5, 1 mL = 3, 0.1 mL = 1; the MPN is 110, and the low and high confidence limits are 31 and 250. For row 42, the reactions are 10 mL = 5, 1 mL = 3, 0.1 mL = 2; the MPN is 140, and the low and high confidence limits are 37 and 340. For row 43, the reactions are 10 mL = 5, 1 mL = 3, 0.1 mL = 3; the MPN is 180, and the low and high confidence limits are 44 and 500.

Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
Theophilus Reply
what's lochia albra
Kizito
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Source:  OpenStax, Microbiology. OpenStax CNX. Nov 01, 2016 Download for free at http://cnx.org/content/col12087/1.4
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