Scientific Notation and Logarithms



Scientific Notation and Logarithms




EXPONENTS AND SCIENTIFIC NOTATION


In mathematics, multiplication is used as a faster method than simple addition when performing multiple addition problems. For example, 6×8=48. The same answer could be found by adding 6 eight times: 6+6+6+6+6+6+6+6=48.


Exponents are a way to simplify complex multiplication problems.


Exponents are written as xa where x is called the base and a is the exponent. The base is the number that is to be multiplied by itself, and the exponent determines how many times it will be multiplied.


For example, 24 is equal to 2 × 2 × 2 × 2 or 16 and 33 is equal to 3 × 3 × 3 or 27. In complicated calculations involving many exponents, it is easy to see how the use of exponents can reduce the chance of errors in performing the arithmetic of the calculation.


In the clinical laboratory, scientific notation is often used in calculations to simplify the calculation. In scientific notation, a number is written in such a way that it is larger than 1 but less than 10 and an integral power of 10. For example, the number 2530000.0 can be expressed as 2.53 × 106 because the decimal point can be moved six places to the left. The 2.53 is referred to as the mantissa number by some mathematicians. This book will also refer to it as the mantissa number.


2563504030201.0=2.53×106


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There are some simple rules for using scientific notation. If these rules are understood and followed, errors in calculations using scientific notation will be reduced.



Rule 1


Exponents used in scientific notation can be positive or negative numbers. Exponents that are negative usually indicate a number that is less than 1. A negative sign is placed to the left of the exponent to indicate that it is a negative exponent. Exponents that are positive generally do not have any sign associated with them. It is assumed that if a negative sign is not present, then the exponent is positive.




Negative Exponents


Numbers with negative exponents are expressed with the following formula:


ba=1ba


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Example 2–2

In this example, from the equation for negative exponents, b = 10 and a = −5.


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Additional Examples


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Notice that with scientific notation, the final product could be easily obtained by simply moving the decimal point to the left for a negative exponent and to the right for a positive exponent the number of times indicated by the exponent number itself. In Example 2–1, 6 × 103 was shown to be equal to 6000. Instead of multiplying 10 by itself three times, simply use 6.0 as the starting point and move the decimal to the right three places.


6.0base60.0onedecimalplace600.0twodecimalplaces6000.0threedecimalplaces


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For a negative exponent, the decimal place is moved to the left. Using 3 × 10-5 from Example 2–2, the decimal point is moved five places to the left.


3.0base0.3onedecimalplace0.03twodecimalplaces0.003threedecimalplaces0.0003fourdecimalplaces0.00003fivedecimalplaces


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Rule 2


If a mantissa number greater than zero has an exponent raised to the zero power, the exponent has a value of 1. This is expressed mathematically as follows:


b×100=b×1=b


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A good understanding of how exponents function in scientific notation may help the reader see why the power of 10 raised to the zero power has a value of 1. The exponent tells the reader how many times 10 must be multiplied by itself. The zero exponent simply means that the power of 10 in scientific notation is to be used zero times or not to be used at all. The mantissa number associated with the power of 10 remains the same: that is, multiplied by 1. Table 2–1 demonstrates powers of 10 used commonly in the clinical laboratory and their values.





Rule 3


When multiplying two mantissa numbers using scientific notation, the mantissa numbers themselves are multiplied but the exponents are added. This rule can be expressed as follows:


[(b×10a)(c×10d)]=(bc)×10a+d


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Rule 5


In a division calculation involving mantissa (a + c) numbers in scientific notation, the exponent in the denominator (b-d) is subtracted from the exponent in the numerator of the equation. This rule is expressed mathematically as follows:


a×10bc×10d=ac×10bd


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Rule 7


When performing addition and subtraction using scientific notation, it is easy to make arithmetic errors. This is because when performing addition and subtraction, the power of 10 exponents used for the scientific notation do not follow the same rules as for multiplication and division. The best way to perform addition and subtraction is to first convert all the numbers in the calculation to their original nonscientific notation form and then perform the addition or subtraction. When using a calculator to perform addition or subtraction involving scientific notation, the EXP (or EE depending on your calculator) function is used. The calculator will convert the numbers to their nonexponent form when performing the calculation.








LOGARITHMS


The inverse of the exponential function y = ax is called a logarithmic function. The value of the logarithmic function is called the logarithm.


In the laboratory, logarithms are expressed in base 10 and are called common logarithms. In simple terms, a logarithm is the exponent. Table 2–2 demonstrates the multiple of 10 relationship between logarithms and exponents. Logarithms are composed of two parts: a characteristic, which is a whole number, and a mantissa, which is the decimal part. The logarithm can be either positive or negative. If it is negative, the number it represents has a value between zero and one. A positive characteristic means that the number the logarithm represents is greater than 1. The log of 1 is equal to zero. Remember that 100 = 1 and that a logarithm is the inverse of the exponent. Table 2–2 is a chart of exponents of 10 and their associated logarithms.





Determining Logarithms


Logarithms can be determined by use of a calculator or by a logarithm table. Most scientific calculators have a display key to calculate logarithms. The number is entered into the calculator, and the display key “log” is pressed to display the logarithm. Because of the ease of use of the scientific calculator in determining logarithms, use of the logarithm table is becoming less common. The logarithm table is in Appendix 2–A and contains the logarithms of numbers between 1.0 and 9.9. To use the logarithm table, find the first two digits in column N, and move to the right for the third digit of the number. The four-digit number in the columns is the logarithm of the number and a decimal is placed to the left of the logarithm.






Example 2–12

Find the logarithm of 16.4. To determine the logarithm of 16.4, follow the steps listed below:



1. Express 16.4 as a number less than 10:16.4 = 1.64 × 10.


2. Determine the logarithm of 1.64 × 10: log 1.64 × 10 = log 1.64 + log10.


3. Determine the logarithm of 1.64 from the logarithm table.


4. Find 1.6 in column N and find the entry to the right of 1.6 under column 4.


5. The four-digit number in column 4 in row 1.6 is .2148.


6. The logarithm of 1.64 is .2148.


7. Find the logarithm of 10 from Table 2–2.


8. The logarithm of 10 = 1.


9. Since the log 1.64 + log 10 = log 16.4, substitute into the equation the appropriate logarithms.


10. 0.2148 + 1.0 = 1.2148


11. The number 1.2148 must be reduced to the same number of significant figures as the original number. The number 16.4 contains three significant figures. Because the rules of significant figures apply only to the mantissa, the mantissa must be rounded from four digits to three.


12. The logarithm for 16.4 = 1.215.




Example 2–13

Find the logarithm of 0.0365. Follow the steps listed below to determine the logarithm of 0.0365:


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Nov 18, 2017 | Posted by in PHARMACY | Comments Off on Scientific Notation and Logarithms

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