Multiplication and Division by Logarithmic Algebra (lig)

In order to solve calculation using logarithmic algebra, each number in an equation is first converted to its logarithmic value.

By definition, the logarithm of a number equals the exact exponent (power) of a base number (10 in the case of scientific numbers) that will produce that number.

Also by definition, the antilogathrim of a number equals the number that results when the base number (10 in the case of scientific numbers) is raised to a power by a logarithm.

For numbers that are pure powers of 10, to conversion to logarithms is obvious. For example, the logarithm of 104 equals 4, or log (104) = 4. The logarithmic conversions for the other numbers in this equation require access to a logarithmic table, except for the number 1 of course, i.e., log (1) = 0, or 100 = 1. However, you may not really need a table because good approximations are possible by knowing approximate logarithmic values for just a few numbers.

Useful Logarithmic Approximations
log (2) = 0.3 (actually 0.301)
log (3) = 0.5 (actually 0.477)
log (5) = 0.7 (actually 0.699)

You should commit these equivalent values to memory!

With these approximate logarithmic values, an approximate solution for the equation presented on the preceding web page -- (2 x 104) x (3 x 10-5) / (4 x 109) -- is found in the following way.

  1. Convert every number in the equation to its logarithmic value.
  2. Add or subtract the logarithms (exponents) according to multiplication or division.
  3. 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 that a couple of steps here were not completely written out here. Namely, the log (4) in the denominator was approximated as follows:

4 = 22 = (100.3)2 = 10(0.3 + 0.3) = 100.6

Also, for 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 approximations above. Therefore,

100.2 = 10(0.5 - 0.3) = 10(0.5)/10(0.3) = 3/2 = 1.5 (approximately).

As illustrated by two acid titration plots for acetic acid , the utility of logarithmic algebra will become more evident when the properties of several reversible, weakly interacting systems are considered, especially in situations where calculations and graphical interpretations are required to make quantitative predictions about the behavior of such systems following changes in one of the interacting components.

Go to top
Return to Biochemistry 108A Home Page
Press the browser's "Back" key to return to previous page.
Introduction to Logarithmic Algebra
Search for a Specific Topic

Duane W. Sears ©
Revised: August 10, 1998