Introduction to
Multiplication
and Division
by Logarithmic Algebra |
|
Calculations involving complex
multiplication
and/or division of numbers, can sometimes be simplified by performing the calculations with
logarithmic algebra. Logarithmic algebra is based on the fact that
powers or exponents of
base numbers are added when
multiplying, or are subtracted when dividing.
- By definition,
the logarithm of a number equals the exact exponent (power) of a base number (10 in the case of scientific
notation) that will produce that
number. For example: log (1.0) = 0.0 since
100.0 =
1.0.
- Also by definition,
the antilogathrim of a number equals the number that results
when the base number (10 in the
case of scientific notation) is raised to a power by a logarithm;
e.g., antilogathrim = 1.0 = 100.0.
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With numbers represented in
scientific notation (i.e., as powers, x, of the base number
10, or 10x), the
exponents of 10 are added when two numbers are
multiplied or are subtracted when the numbers are divided, as illustrated by the example below:
| (2 x 104) x (3 x 10-5)
/ (4 x 109) = (2 x 3 / 4) x 10(4-5-9) = 1.5 x 10-10 |
For numbers that are pure powers of
10, the conversion to logarithmic values is easy. For example, the log ( 104) equals
4. For most numbers, however, logarithmic conversions require access to
a logarithmic table. However, a log table is not really needed in order to approximate
most calculations if the following approximate log values are employed.
Table
of Useful Logarithmic
Approximations
log
(1) = 0.0
log (2) = 0.3 (actually 0.301)
log (3) = 0.5 (actually 0.477)
log (5) = 0.7 (actually 0.699)
log (7) = 0.85 (actually 0.845) |
With these approximate logarithmic values, approximate
solutions for most multiplication products and division quotients can be
determined. For example, consider the calculation, (2 x 104) x (3
x 10-5) / (4 x 109):
- Convert every number in the equation to its logarithmic
value using the table above.
- Add or subtract the logarithms
(exponents) according to multiplication or division.
- Determine the final calculated answer by taking the antilogarithm of the composite sum of exponents.
(2 x 104) x (3 x 10-5)
/ (4 x 109)
= (100.3 x 104) x (100.5 x 10-5)
/ (100.6 x 109)
= 10(0.3
+ 4 + 0.5 -5 - 0.6 - 9) = 10(-9.8) = 10(-0.8) x 10(-9)
= 100.2 x 10-10 = 1.5 x 10-10 |
Note: a few steps were not completely
elaborated. Namely, log (4) in the denominator was found as follows:
4 = 22 = (100.3)2 = 10(0.3
+ 0.3) = 100.6
Also, in the last step, the antilogarithm of 100.2
was approximated by "working backwards" and noting that log (3) - log (2) = 0.2
from the table of log approximations above. Thus,
100.2 = 10(0.5 - 0.3) = 10(0.5)/10(0.3)
= 3/2 = 1.5.
