Introduction to Logarithmic Algebra (lig)

Numerical values are typically converted to their corresponding logarithmic values in the following situations:

• Functions that range over a broad scale of magnitudes are usually graphed with a logarithmic axis (or axes).
• Calculations involving the multiplication and/or division of combinations of very large and/or very small numbers are sometimes simpler with logarithmic algebra.

The utility of logarithmic scales is particularly evident when comparing numbers or values that are disproptionately different in magnitude, as illustrated by a portrait of our President comapred to the Washington Monument .

Calculations involving complex multiplications and/or divisions, 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 and subtracted when dividing. For example, with numbers represented by scientific notation (i.e., as powers, x, of the base number 10, or 10^x), the exponents of 10 are added when two numbers are multiplied and subtracted when divided, as illustrated by the example below:

To perform the same calculation using logarithmic algebra , every number in the equation is first converted to its logathrimic value before the logathrims (exponents) are added or subtracted together and the final calculated answer is found by taking the antilogathrim of this composit sum.

When data is plotted on a logarithmic scale, the numbers are typically represented in one of two ways.

1. The data numbers themselves are plotted on a logarithmic axis.
2. The data numbers are converted to their logarithmic values (exponents of 10) which are then plotted on a linear axis.
   

The two scales above cover an identical range of numbers but differ in their utility as illustrated by two representations of the titration curve of acetic acid .

The main advantage of the logarithmic scale, in either of its forms above, is that it retains the exponent additivity of any over any range of numbers as demonstrated by some examples of how to use a slide rule .

• Click here to return to the Critical Concepts Web for Logarithms .

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Duane W. Sears ©
Revised: August 10, 1998