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MATH 99 LOGARITHM BASICS Logarithms were developed to allow multiplication, division, and exponentiation of very large and very small numbers, before the advent of the electronic calculator. Mathematically the need to perform these calculations using logarithms has been eliminated by the use of the electronic calculator. Logarithms are a convenient way to represent very small & large numbers with very few digits. They are used in areas such as: electronics - decibels, seismology - Richter Scale, Chemistry - pH A logarithm of a number is a power, that is, it is the exponent that a number called the base must be raised to get the original number. This is best said by example. We will use the base two (2). The logarithm of 8 (to the base 2) is 3, since: 23 = 8. Mathematically this is written: log2 8 = 3. What is the log2 64 = ? It is 6 since: 26 = 64. What is the log 2 1 1 1 ? It is -4 since: 2-4 = 4 16 16 2 The following table lists numbers & their logarithm to the base 2. Number N 2n 1 8 2-3 = (inv-log) log2 N 1 4 1 23 -3 2-2 = -2 1 2 1 22 2-1 = -1 1 21 1 2 4 8 20 21 22 23 0 1 2 3 MULTIPLICATION OF NUMBERS BY ADDITION OF THEIR LOGARITHMS 1) Consider the following multiplication: 2 8 It is equivalent to: 21 23 Rules for multiplication with exponents yields: 21 + 3 = 24 = 16 Since logarithms are exponents, multiplication of numbers is equivalent to adding their logarithms: log2 2 = 1 & log2 8 = 3 1 + 3 = 4, which is the exponent 2 must be raised to in order to get the result from the multiplication. Raising 2 to this power of 4 is called taking the inverse-log or anti-log of 4. Taking the inv-log2 of 4 results in the solution. inv-log 4 24 16 2) Multiply: 4 1 8 What is the log2 4 = ___________ & log2 (1/8) = __________ What is the sum of the logs? ________________ Page 1 of 2 File: LogBasic.DOC Revised: 6/28/2017 What is the resultant of the multiplication, i.e. the inv-log2 2sum of logs = __________ SOLUTIONS: 2; -3; -1; 1/2 DIVISION OF NUMBERS BY SUBTRACTION OF THEIR LOGARITHMS 1) Consider the following division: 8 (1/4) = 8 (4/1) = 8 4 = 32 It is equivalent to: 23 2-2 The rules for division with exponents yields: 2[3- (-2)] = 25 = 32 Since logarithms are exponents, division of numbers is equivalent to subtracting their logarithms: log2 8 = 3 & log2 (1/4) = -2 3 - (-2) = 5, which is the exponent 2 must be raised to in order to get the result from the division. Raising 2 to this power of 5 is called taking the inverse-log or anti-log of 5. Taking the inv-log2 of 5 results in the solution. inv-log2 5 25 32 2) Divide: 1 8 2 What is the log2 (1/2) = ____________ & log2 8 = _____________ What is the difference of the logs? _______________________ What is the result of the division, i.e. the inv-log2 2difference of logs = _______________ SOLUTIONS: -1; 3; -4; 1/16 RAISING NUMBERS TO POWERS BY MULTIPLICATION OF THE POWER & THE LOG OF THE NUMBER 1) Consider the following number raised to the power: 43 = 64 It is equivalent to: 444 = 41 41 41 = (41)3 Rules for the multiplication with exponents yields: 4(1+1+1) = 43 = 64 Since logarithms are exponents, multiplication of numbers is equivalent to adding their logarithms: log2 4 = 2 2 + 2 + 2 = 6, which is the exponent 2 must be raised to in order to get the result of the multiplication. The exponent 6 can also be the result of 3(log2 4) = 3(2) = 6 Raising 2 to this power of 6 is called taking the inverse log or anti-log of 6. Taking the inv-log2 of 6 results in the solution. inv-log2 6 26 64 2) Raise to a power: 83 What is the log2 8 = ___________ What is 3 times this value i.e. 3(log2 8) = 3log2 8? = ______________ What is the result of the raising to the power (exponentiation), i.e. inv-log2 23log = ________________________ SOLUTIONS: 3; 9; 512 Page 2 of 2 File: LogBasic.DOC Revised: 6/28/2017