THE RULES OF LOGARITHMIC COMPUTATION

 

In the first worksheet the rules of logarithmic computation are introduced. With the aid of a computer algebra system like DERIVE, students try to discover the three main rules of logarithmic computation, namely addition and subtraction of logarithms and calculating logarithms of the power function.


 


We should also warn the students that in DERIVE log (x) does not denote a common logarithm or a Briggsian one as it is usual. Logarithmic function with the basis a is written as log (x, a) and logarithmic function with the basis e with log x or ln x. As DERIVE transforms logarithms with other bases to natural logarithms and thus prevents us from building up too many rules, natural logarithms are used in the worksheet.

 

Figure 1

 

Also the teacher should check whether DERIVE is appropriately set up. Namely, the simplification settings for Logarithmic transformations should be set to auto. This ensures that rules

 

 

are used by DERIVE. This can be checked in Declare/Algebra state/Simplification (DERIVE for Windows 4) or with  Declare/Simplifica-tion settings (DERIVE for Windows 5)

 

 

Figure 2

 

As default settings are such, we do not include this explanation into the student's worksheet. Also, if we modify the exercises, we should be careful, especially with subtraction. Namely if the second number is bigger than the first one we get the result like in expression #2 in the Figure 1. Also if the result of applied transformation can be additionally simplified, DERIVE does that too and that might confuse the student.

We do not expect students to have any difficulties with the mathematical notation of rules, but more problems will probably occur when the rules have to be expressed with words and suitable sentences made. We think this part is the important one; so we should insist in solving this part too, regardless of the time, which is spent on clarifying this question. The whole exercise is quite short, so a substantial amount of time can be devoted to discussion. We also direct students to check various WEB resources devoted to logarithmic function, such as:

 

On history of logarithmic function:

 

 

On Eulers' number e and natural logarithm:

 

 

On rules for calculating with logarithms:

 

 

and many more, especially on Ask Dr. Math:

 

 http://forum.swarthmore.edu/dr.math/tocs/logarithm.high.html

 

In the next lesson the classical way of teaching is assumed to be used where the rules that the students had discovered before are also proven in a mathematically correct way.

 

For solving the exercise we assume no prior knowledge of the DERIVE program, so all necessary steps with entering the expressions and similar are described. Since all computations will be done with natural numbers only, DERIVE should be set appropriately, namely so that all variables are assumed to be natural numbers.

At the moment most of our schools have Derive for Windows Version 4, but as the new version, Version 5, seems to be much more appropriate for school use, both versions are explained whenever necessary.

The electronic form of the students' worksheet can be found on

 

http://rc.fmf.uni-lj.si/matija/logarithm/worksheets/rules.htm