|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
course syllabus assignments materials request a test proctor form |
Multiplying by Addition!Historically, common logs (base 10) were very helpful in performing, by hand with a pencil and paper, multiplication (and division) because using logs transformed these lengthy operations into addition (and subtraction) which are faster to do. The use of common logs and scientific notation meant that we would only need a restricted Log Table. The mathematical relationship we use is one of the properties of the logarithmic function:
log xy = log x + log y Of course, we really want the product, xy, not its log. We could get this by exponentiating both sides of the equation using the base 10. However, this can be done with the same table! If the entries in the body of the table are the logarithms of the indices, then the indices are the base 10 exponents of the values. Using log tables, this process is termed the antilog and is accomplished by looking in the body of the table for the value then reading off the index that corresponds to it. Thus: xy = antilog(log x + log y) where the the two log functions are implemented by finding the argument in the index of the table and writing down the corresponding values. The antilog function is performed by a "reverse" lookup. It might seem that we would need a very large Log Table to contain all possible numbers. However, if we write the original numbers in scientific notation, then the needed size of our table is quite small. Scientific notation writes every value as: d.ddd...×10p The "value" part, d.ddd..., is called the mantissa and is always in the interval [1, 10); that is, it is written so that there is exactly one leading non-zero digit (to the left of the decimal point). The exponent of 10 is the number of places the original decimal point must be moved to obtain this. As examples: 34,512 become 3.4512×104 and 0.00256 is equal to 2.56×10−3 Further, the log of such a value can be written as: log 3.4512×104 = log 3.4512 + log 104 = log 3.4512 + 4 That is, we can easily determine the log of the exponent. We only need to look up the log of the mantissa which is from 1 to 10 (actually 9.9999999...). Because the mantissa is always from 1 to 10, we only need a Log Table with entries from 1 up to 10! As an example let us calculate: 5.372×104 times 2.743×106 First we need to find the logarithm of the first value: log (5.372×104) = log 5.372 + log 104 = log 5.372 + 4 We need to find the logarithm of 5.372. To do this we use a Log Table. The table we will utilize is organized with the first three digits on the left and the fourth digit across the rows. This means this table has 999 rows with 10 values in each row! The relevant portion of the Log Table would appear as:
From the table (the shaded entry), we find: log 5.372 = 0.73014 Thus log 5.372×104 = 4.73014 Now we need to repeat this process for the second value: log (2.743×106) = log 2.743 + log 106 = log 2.743 + 6 The relevant portion of the Log Table is:
From the table (the shaded entry), we find: log 2.743 = 0.43823 Thus log 2.743×106 = 6.43823 Therefore log (5.32×104•2.73×106) = 4.73014 + 6.43838 = 11.16837 This can be written as: 11 + 0.16837 This is the log of the answer. As we see from the pattern above, the 11 corresponds to a 1011 factor and the 0.16837 corresponds to the logarithm of the "value" factor. To find what the value is, we look in the body of the Log Table for an entry that is closest to 0.16837 and then determine what number (on the left and top) has this as a logarithm:
The value 0.16837 falls between 0.16820 (which corresponds to 1.473) and 0.16850 (which corresponds to 1.474). Therefore, we obtain as a "almost" final answer: 1.473×1011 Actually, we get a little better. We can use a linear interpolation to estimate the fraction between 1.473 and 1.474. We take the difference between the actual value and the next lower value in the table and divide this by the difference between the two adjacent table values. In this case we get:
beyond the digits we already have to get as a final answer:
1.4736×1011 The actual value is:
1.4735396×1011 While this proceedure seems cumbersome and a lot of work, it much faster and less work than multiplying the two numbers when you have to perform the calculations "by hand!" If we had a Log Table with more digits, then our computed answer would be closer to the true value. If we have a negative exponent in one or both of our original numbers, then we cannot simply catenate the integer and the fraction. We really need to subtract them to the correct log. As an example the log of 3.2×10−4 becomes −4 + 0.50515 = −3.49485 If the logarithm of the result is negative, we similarly have one further step before performing our "reverse" lookup. When the result is negative, both the integer part and the fractional part are negative. However, to use the Log Table, the fractional part needs to be positive. We need to subtract 1 from the integer part and subtract the fractional part from 1.0000. This makes the fractional part positive in the proper range [0,1). To see how this works, consider the value of a logarithm that is −3.25. This is equivalent to −3 + −0.25 which in turn is equivalent to −4 + 0.75. The 0.75 corresponds to 5.6234. Therefore, the antilog or actual value is 5.6234x10−4 or 0.00056234. We could perform division simply by subtracting the logs of the two values instead of adding using the logarithmic relation:
x/y = antilog(log x − log y)
We can raise a value to a power by using the relationship: log xp = p log x or xp = antilog(p log x) |
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
© 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 by Lawrence Turner |
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||