Math 151
Fall 03
Logarithms
The “log”
button on your calculator finds the base 10 logarithm of a number. If ,
then
.
1. Find the (base 10) logarithms of 10, 100, 1000, 10,000 etc.
Find the logarithms of 0.1, 0.01, 0.001, 0.0001, etc.
What patterns do you notice? Reconcile your patterns with the definition of logarithms.
2. Find the base 10 logarithms of 2, 20, 200, 2000, 20,000 0.2, 0.02, 0.002 etc.
What patterns do you notice? Try some other numbers and see whether your patterns hold for them too. Make a conjecture (or several) based on your pattern(s).
3. Find and repeat for other examples you’ve
tried. Make a conjecture about the value
of
. Use the definition of logarithms to confirm
or deny your conjecture.
4. Show that . Reconcile with earlier conjectures.
5. Show that . Hint: let
and
. Relate to patterns you found earlier.
Note that the property in this exercise is the key property
of logarithms. Before calculators and
computers, people used logarithms to reduce multiplying large numbers to
addition problems. Slide rules work
because of logarithms (if you don’t know what a slide rule is, look it up not as ancient as you might think! This author
learned to use one in high school in the ‘70’s).
6. Show that ;
hint: let
,
and use the previous exercise. Find
similar expressions for
,
etc. Make up a conjecture about the
value of
.
7. Now we will use
all of our results so far to solve the equation we found in computing the
fractal dimension of a Sierpinski Gasket, namely . By taking logarithms of each side and using
your results from #6, show that
.
Copyright 2005, Debra K. Borkovitz. You may copy or edit this material for non-profit, educational use only.
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