LOGARITHMIC FUNCTIONS
The concept of logarithmic functions is a subject of great confusion when first encountered by the student in the classroom. Perhaps that difficulty is caused because the subject is considered by instructor and student alike as a form of reality, rather than as an imaginary mathematical game.
The ultimate simplicity becomes apparent when it is recognized that word logarithm is simply another word of identical meaning to the solution of a mathematical equation.
There are two primary parts to the solution of a mathematical equation which is referred to as a logarithm. These two parts of the equation is named the ‘base’ and the ‘exponent’. These are simply code words which refer to one number, the ‘base’, which will be multiplied by itself by some other number of times referred to as the ‘exponent’. For example, the mathematical equation displayed as 10^3 is simply the base number of 10 multiplied by itself three times or 10 x 10 x 10 = 1000. In effect, the value of 1000 is a solution to a problem - but that solution is then given another code name of ‘logarithm’.
After that solution of 10^3 has been determined to mean 1000, and renamed as a ‘logarithm’. we are then advised that ‘logarithm’ of 1000 to the base number of 10 is 3.
At the time that I graduated from engineering school (many years ago) that utter simplicity about the meaning of the word ‘logarithm’ had never yet occurred to me I simply did not understand that the word ‘logarithm’ was another name for the solution of a strange exponential mathematical equation.
I am not aware of the history involving the how and why the special vocabulary pertaining to the solution to exponential mathematical functions was developed. But there is one possibility that relates back to another imaginary mathematical process referred to as ‘calculus’. The process referred to as calculus was investigated in the prior section of this document.
That process of calculus seemed to ‘work’ when the function of X (shown as f (X)) involved exponential values of X greater than 1.0. However, when the exponential values were equal to, or less than 1.0, the normal process of calculus failed. For example when the function of X is (1/X) then the graph of that function is asymptotic to the coordinates of the graph of X versus f(X).
This is depicted here where the horizontal and vertical axis of the graph are located at the center of an imaginary coordinate system located at a point referred to as ( 0, 0 ), and values along both coordinate lines increase to infinity.
When the value of the exponent factor is equal to 1.0, then the value of the resultant logarithm will be equal to the base number. As the value of the exponent factor increases above 1.0, the resultant solution value increases ever more rapidly until it approached infinity. However, when the value of the exponent decreases below 1.0, then the value of the solution decreases ever more slowly towards zero. The final values of f(X) equal infinity and zero are never actually reached, as the curve is asymptotic to the axis. In order to determine the integration process of the f(X) function, between X= 0 and any other limit, it is apparent that the area of interest must be equal to the imaginary value of ‘infinity’. And since that imaginary value is not acceptable, the mathematical process named ‘calculus’ fails for this function. Perhaps it was at this point that the concept of ‘logarithms’ was created as a means to resolve the failure of calculus when an attempt to apply the normal rules of calculus to the function of 1/X failed.
Interestingly, the rules of mathematics have been declared to demand that when the exponent has a value less than 1, then that exponent becomes the denominator of a fractional value, and/or can be expressed as a negative value rather than a fraction. Which is to say that when f(X) = X^N and N has a value less than zero, then f(X) is by definition revised from X^(-N) to 1/ X^(+N) . I suppose there may be some sort of logical explanation for this strange rule, but perhaps the rule was established purely by establishment of a new set of definitions needed to explain the failure of calculus to ‘work’ when applied to the family of functions of X^(-N).
In any event, after the rules were applied, and the vocabulary of logarithms had been created, then the resultant graph for logarithmic functions appeared as indicated by this graph:
Of special interest in this presentation is that the concept of fractional values between zero and 1.0 has been modified to a concept where the distance between zero and 1 has become equal to an infinite range of values which may be displayed - but do not actually represent, an infinite range of values extending from zero to minus infinity. That reality is disguised because, by definition only, all real values from one to zero are now represented by the denominator of a fraction represented as 1/X.
If we relocate the coordinate axis to the location where the exponent has a value of 1, and change the sign of the exponent so that it increases negatively to the left of the vertical coordinate, then the same information appears as shown here. The values on the vertical axis will increase at a rate X times faster than the values of the linear values that are indicated horizontal axis.
It is this format of graph that is referred to as a ‘semi-log’ graph representation format. Typically, graph paper utilized has vertical grid marks that are equally spaced but labeled as representing values that increase by an orders of magnitude of ten.
Back to Calculus
Onward to Geometric Functions
return to DIALOG page
Home Index