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 x^a 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.

25︸63︸50︸40︸30︸20︸1.0=2.53×106

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.

This problem can be solved two different ways: From Table 2–1, 10⁰ = 1. Substituting this equivalent value into the equation yields 3.8 × 1, which equals 3.8. From Table 2–1, a × 10⁰ = a, or using the example, 5 × 10⁰ = 5. By substituting the numbers in the problem into the equation, 3.8 × 10⁰ = 3.8.

Additional Examples

What is 8.9 × 10⁰?

Using the table above or the example above, then 8.9 × 10⁰ = 8.9.

What is 7.4 × 10⁰?

Using the table above or the examples above, then 7.4 × 10⁰ = 7.4.

What is 5.3 × 10⁰?

Using the table above, or the examples above, then 5.3 × 10⁰ = 5.3.

Exponent	Mantissa Number (a) and Exponent	Value	Example
10⁰ = 1	a × 10⁰	a	5 × 10⁰ =5
10¹ = 10	a × 10¹	a × 10	5 × 10¹ =50
10² = 100	a × 10²	a × 100	5 × 10² =500
10³ = 1000	a × 10³	a × 1000	5 × 10³ =500
10⁶ = 1,000,000	a × 10⁶	a × 1,000,000	5 × 10⁶ =5,000, 000
10⁹ = 1,000,000,000	a × 10⁹	a × 1,000,000,000	5 × 10⁹ =5,000,000,000
10^-1 = 0.1	a × 10^-1	a × 0.1	5 × 10^-1 = 0.5
10^-2 = 0.01	a × 10^-2	a × 0.01	5 × 10^-2 = 0.05
10^-3 = 0.001	a × 10^-3	a × 0.001	5 × 10^-3 = 0.005
10^-6 = 0.000001	a × 10^-6	a × 0.000001	5 × 10^-6 = 0.000005
10^-9 = 0.000000001	a × 10^-9	a × 0.000000001	5 × 10^-9 = 0.000000005

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

Example 2–4

Multiply 4.5 × 10² by 4.9 × 10³. Using Rule 3, the following equation is derived:

This equation can be verified by using simple arithmetic:

4.5×102=4504.9×103=4900450×4900=2205000

The difference between 2205000 and 2.2 × 10⁶ is because of significant figures rules of calculations. The number 2205000 is rounded to two significant figures.

Additional Examples

Multiply 2.15 × 10² by 6.72 × 10².

Using Rule 3, the answer is 1.44 × 10⁵.

Multiply 9.3 × 10³ by 4.5 × 10².

Using Rule 3, the answer is 4.18 × 10⁶.

Multiply 3.6 × 10⁴ by 1.3 × 10².

Using Rule 3, the answer is 4.68 × 10⁶.

Rule 4

When a mantissa (a) number in scientific notation is multiplied by an exponent, the mantissa (a^c) number is multiplied by itself the number of times expressed by the exponent. The power of 10 exponent is multiplied by the exponent. This is expressed mathematically as follows:

(a×10b)c=ac×10bc

a^c is equal to “a” multiplied by itself “c” times. This is expressed mathematically as

a^c = a × a × a…(c times)

Example 2–5

Additional Examples

(8.7 × 10³)² = 8.7² × 10^3×2 = 75.6 × 10⁶ = 7.56 × 10⁷ = 7.6 × 10⁷

(1.4 × 10²)² = 1.4² × 10^2×2 = 1.96 × 10⁴ = 2.0 × 10⁴

(3.2 × 10²)² = 3.2² × 10^2×2 = 10.24 × 10⁴ = 1.0 × 10⁵

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×10b−d

Example 2–6

Divide 7.95 ×10³ by 2.5 × 10². Using the formula from Rule 5:

This answer can be confirmed by simple arithmetic:

Additional Examples

Divide 5.2 × 10³ by 2.7 × 10² = 1.9 × 10¹.

Divide 1.6 × 10² by 4.8 × 10¹ = 0.33 × 10¹ or 3.3 × 10⁰ or 3.3.

Divide 9.3 × 10⁴ by 8.4 × 10³ = 1.1 × 10¹.

All calculations performed using scientific notation should arrive at the same answer that could be obtained by the slower simple arithmetic method. It is a good idea to check your answers while becoming familiar with scientific notation to make sure that mistakes are not made.

Rule 6

A different but comparable way of expressing a division problem is the rule that in a division problem the exponent in the numerator can be subtracted from the exponent in the denominator. This is expressed mathematically as follows:

a×10bc×10d=ac×110d−b

Example 2–7

The same calculation from Example 2–6 will be used to demonstrate Rule 6. Using the formula from Rule 6:

When a negative exponent is in the denominator of an equation, it becomes a positive exponent when inverted to the numerator. This is why 110−1 became a positive 10¹.

Additional Examples

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.

Example 2–8

Add 5.23 × 10⁴ + 9.2 × 10².

If we try to manually solve this problem by using the same exponent rule as for multiplication, then the equation that will be derived is as follows:

(5.23+9.2)×106

Will this equation result in the correct answer? NO! Logic tells us that this equation cannot result in the correct answer. If, however, we first convert each number that is in scientific notation back to its simpler form and then perform the addition, we would expect to arrive at the following:

Additional Examples

3.81 × 10⁵ + 5.98 × 10⁴ = ?

Solving it manually: 381000 + 59800 = 440800 or 4.41 × 10⁵.

7.7 × 10² + 5.5 × 10³ = ?

Solving it manually: 770 + 5500 = 6270 or 6.3 × 10³.

Example 2–9

Subtract 2.3 × 10² from 7.5 × 10³.

In subtraction, there is no easy way to perform this calculation without a good chance of error except by first converting all terms into their simpler forms if the subtraction is performed manually or by using a calculator and the EXP (or EE) function.

Up to this point, all the examples have used positive exponents. The same rules apply when working with negative exponents. A common mistake when performing calculations involving negative exponents is to forget that when a negative number is subtracted from a positive number, it is actually added to the positive number because of the following rule that was discussed in Chapter 1:

Two Negatives Rule

Two negatives become a positive.

Example 2–10

5.0 × 10^-2 divided by 6.0 × 10⁻³ = ? Using the rules of division with exponents, the following equation is derived:

When −3 is subtracted from −2, it is actually added to −2 and so becomes a positive number (+1) or [(−2) − (−3) = −2 +3 = +1].

To confirm these answers, convert all numbers to their original nonscientific notation value:

Additional Examples

LOGARITHMS

The inverse of the exponential function y = a^x 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.

TABLE 2–2

Exponents and Their Associated Logarithms

Scientific Notation	Exponent	Logarithm
1 = 10⁰	0	log 1 = 0
10 = 10¹	1	log 10 = 1.00
100 = 10²	2	log 100 = 2.000
1000 = 10³	3	log 1000 = 3.0000
10000 = 10⁴	4	log 10000 = 4.00000
100000 = 10⁵	5	log 100000 = 5.000000
1000000 = 10⁶	6	log 1000000 = 6.0000000
0.000001 = 10^-6	−6	log 0.000001 = -6
0.00001 = 10^-5	−5	log 0.00001 = -5
0.0001 = 10^-4	−4	log 0.0001 = -4
0.001 = 10^-3	−3	log 0.001 = -3
0.01 = 10^-2	−2	log 0.01 = -2
0.1 = 10^-1	−1	log 0.1 = -1.0