http student ccbcmd edu cyau1 LogarithmSp99 htm Brief Review of Logarithm Logarithm is the exponent to which a number is raised to produce a given number The common logarithm commonly abbreviated as log refers to the exponent to which 10 has to be raised and the natural logarithm abbreviated ln refers to the exponent to which e has to be raised We will consider only the common logarithm here For example given the number 100 log of 100 is 2 since 10 has to be raised to a power of 2 to produce 100 In the case of log 0 01 the answer is 2 since 0 01 10 2 and we must raise 10 to the power of 2 to produce 0 01 When we want to find the log of a number we are asking what is x in 10x to produce this number See below for more examples Logarithm of Numbers with Exponents that are Integers Log of numbers with exponents that are integers i e numbers that are simple powers of ten do not require the use of calculators or log tables to compute First express the number in exponential form and the log of the number is simply the exponent See examples shown in the table below log 100 log 10 2 2 log 10 log 10 1 1 log 1 log 10 0 0 log 0 1 log 10 1 1 log 0 01 log 10 2 2 log 0 00001 log 10 5 5 Note that when a number increases by powers of ten such as from 10 to 100 the logarithm of these numbers increase only by one unit from 1 to 2 This is referred to as a logarithmic function Certain measurements in chemistry are related to each other by such logarithmic functions e g transmittance vs absorbance H3O vs pH Antilog of Integers Antilog refers to the inverse function of log It can also be written as log 1 To find the antilog of a number we simply reverse the process It is the value equivalent to 10 brought to the power of the given number In other words antilog of 2 is 102 or 100 The antilog of 2 is 10 2 or 0 01 Again this does not require a calculator or a log table to compute See examples shown in the table below antilog 2 10 2 100 antilog 1 10 1 10 antilog 0 10 0 1 antilog 1 10 1 0 1 antilog 5 10 5 0 00001 Since the antilog function is the inverse of the log function the antilog of the log of a number is the number itself For example antilog log 100 100 Logarithm of Numbers with Exponents that are Non integers Logarithm of more complex numbers require the use of a calculator or a log table Only the use of a calculator will be discussed here If you had to calculate the log of a number just enter the number and then press log For example to find log 2 5 enter 2 5 then press log The answer is 0 397940009 For certain calculators such as graphics calculators you need to press log first followed by the number in parenthesis sign the number followed by enter For such calculators it is wise to get in the habit of always putting the number within the parenthesis signs In other words to find log 2 5 it is best to press log 2 5 enter Although it makes no difference in this case it can cause trouble in other situations if the parentheses were omitted See below for example If you want to find log3x5 which has the answer 1 17609 and you pressed log 3 x 5 enter you would have computed log 3 x 5 instead giving you the answer of 2 3856 Especially if you are one of those who insist on entering numbers like 2x103 by pressing 2 x 10x 3 instead of the recommended keystrokes of 2 EE 3 you will have to think very carefully about the order of operation The calculator will do the log function before multiplication and division So if you want to find log 2x103 with the answer of 3 3010 and pressed log 2 x 10x 3 enter you would have computed for log 2 x 103 instead giving you the answer of 301 0299 The problem is avoided by enclosing your number within the parentheses signs Practice by finding the log of the numbers in the table below log 4 2 0 62324929 log 4 2x102 2 62324929 log 4 2x105 5 62324929 log 4 2x10 1 0 37675071 log 4 2x10 2 1 37675071 Significant Figures When Finding the Log of a Number Before we begin let us establish the vocabulary involving exponential expressions and logarithms In an exponential expression such as 4 2 x 10 2 4 2 is called the coefficient and 2 is called the exponent Thus 2 x 103 would have a coefficient of 2 and an exponent of 3 The log of a number consists of two parts The number to the left of the decimal point is called the characteristic and the number to the right is the mantissa Thus the number 3 278 has a characteristic of 3 and a mantissa of 278 The number 3 278 has a characteristic of 3 Let us examine the table again log 4 2 0 62324929 log 4 2x102 2 62324929 log 4 2x105 5 62324929 log 4 2x10 1 0 37675071 log 4 2x10 2 1 37675071 As you can see in the first three examples in the table the mantissa is established by the coefficient 4 2 and the characteristic is established by the exponent In the last two examples this may not be as apparent but log 4 2x10 1 is calculated as log 4 2 log 10 1 which is equal to 0 6232 1 0 37675071 The point is the significant figures of a number should be reflected only in the mantissa of its log In other words the number of significant figures of a number is the number of decimal places in its log CAUTION To figure out the number of decimal places the number MUST be first expressed in decimal form rather than in exponential form For example 4 2 has 2 sig fig so log 4 2 should have only 2 decimal places 0 62 4 25 has 3 sig fig so log 4 25 should have 3 decimal places 0 628 0 03 has 1 sig fig so log 0 03 should have 1 decimal place 1 5 1 01 has 3 sig fig so log 1 01 should have 3 decimal places 4x10 3 0 004 Practice with the examples below log 4 2 0 62 log 4 2x102 2 62 log 4 2x105 5 62 log 4 2x10 1 0 37 log 0 002735 2 5630 log 3200 3 50 log 50000 4 7 log 1 002 9x10 4 If you had trouble with the last 4 examples you need to review the rules for significant figures Determining the Antilog of Numbers that …