Q&A #3572

Logarithmic functions

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Apr 03, 2000 at 17:25:21
Subject: Re: Logarithmic functions

Let me echo Claudia's response, and add a small postscript to it.  The
relationship between logs and exponentials is the most important idea you
will work with in this area.  If they memorize the rules and do not leave your
class knowing how to work back and forth between log(base a) of b = c and
a^c=b then they have gained little of mathematical use.

In the old days we used to have slide rules which provided a physical
analogy for logs and the addition property of exponents, but now you have to
recreate this.  It is certainly worth while to take at least a 15 minute
segment and show them how logs of any base can be constructed with equal
length scales.  Take a foot ruler or some similar object and mark off a
number line on the board and label the units 0,1,2,3  etc... on top of the
tic mark dividers.  Then under the tics write the powers of some base a
(usually I do 2 to start and then replay it with 3 or 5)  2^0=1 goes under 0,
2^1 under 1 etc, and the bottom will look like 1,2,4,8,16...
   Now you can show them that to multiply 16 times 32, we only have to look
at the top where log16=4 and log 32 = 5 and we add 4+5 to get 9 and our
answer, under the nine is 2^9=512. The division rule is just as easy to
illustrate.   It is easy for students to see that the logarithm is just about
the powers of the numbers.

Students quickly realize how easy it is to estimate numbers on the exponent
line (its linear) and how difficult on the non-linear section.  Since the log
of a square root is 1/2 the log of the number, we can bisect powers to locate
some of the irrational results.  Ask students to guess where 25 would be (and
what log(base 2) of 25 is, then by taking 1/2 the log distance we have found
an estimate for 5 (and of log(base 2) of 5.

Changing the base ONLY changes one of the number lines, reinforcing that the
rules of logarithms are independent of the base used... a nice introduction
to why ln can be so easily used to replace the others.

After all this, expect them to still be confused, and still make silly
mistakes.  Honors kids are still, in the main, kids.  They have limited
experience and need time to revisit these ideas after some percolation has
occurred.  Look for opportunities to reinforce the ideas of exponents as you
encounter different topics through the rest of the year.

Good luck.

 -Pat Ballew, for the Teacher2Teacher service

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