in a time before calculators
19/05/2012 § 2 Comments
The current unit we are studying in class is this fabulous thing called logarithms.
log•a•rithm – (n.) a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.
In simpler words, the log of a number is the power (x) that makes the base number (y) equal n. Since that actually wasn’t any simpler than the dictionary definition, let’s put that in actual equations.
(Way more happening under the cut.)
Typically, if the logarithm is difficult and complex enough, students and mathematicians today have that wonderful thing called a calculator to help find the value of a crazy logarithm like log1738 (which is 1.283 – easily found on the calculator).
One would wonder, however, what mathematicians and students back in the 1700s or 1800s would use to find logarithms and to solve complex problems. They didn’t have calculators, they only had their minds.
What they did back then was rewrite the complex equation in terms of logarithms in base 10 (aka log10 n). Because tables and charts of the values of logarithms in base 10 exist, we can just plug in those values into the rewritten equation and use the two laws of logs to further simplify the equation.
The rules of logs are as follows:
- multiplied or divided logs can be changed and should be changed into added or subtracted logs
- exponents and roots in logs can be changed and should be changed into multiplying or dividing values
Now it’s time for an example. (Otherwise, nothing I’d be saying right now would be proven legit and my efforts to explain logarithms would have gone to waste.)
e.g. let’s use this slightly scary but not too terrifying equation 4.834 ÷ 3√5162, and let’s do things step-by-step just so we don’t forget even the slightest alteration:
The image below is my checking on the TI-calculator. The answers correspond with each other, proving my methods and mathematical workings correct and accurate. (Yes, please tilt your head 90° to the left in order to read the calculator.)
I need to be totally honest right now and admit straight out that I made mistakes in the basic math of my equation. What happened was that in steps #7 and #8, I was supposed to do the distributive property and distribute the negative ⅔, making two negative expressions. I was so caught up in getting the same final answer as the calculator that I completely forgot about it and altered the entire answer, getting something totally different. My brother and I were able to locate that problem together and I was able to fix my mistakes.
As for using logarithms, it was strangely comforting to pull apart large and intimidating numbers and forcing them to become small decimalized numbers instead of numbers in the hundreds. When the calculator was not available in this journal assignment, the act of getting a certain number and referring to a table seemed really old-fashioned and methodized. Despite this, however, I thought that it puts further understanding into the student as they see where the numbers come from and also because they dig into the very heart of base 10 logarithms and plug the values in themselves.
I can understand how people did this before the invention of smart calculators. Since the first calculators that only did simple math weren’t invented until the 1960s, the millions of mathematicians and students prior to the release of calculators probably had to do all of these steps and refer to the base 10 logarithm table. Because of this, accuracy may have been more of a problem in the past. There are more steps involved, therefore more chances that someone might make a sloppy mistake, unless of course, they were a math wiz and did all kinds of algebra perfectly. My results were accurate because I spent a lot of time doing the right steps and getting the right answer. I imagine that students back then may not have gotten the amount of time that I have, for example during a test or in class, and wouldn’t have as much time to check accuracy.
Obviously from this study, we can conclude that calculators are great inventions. Although this method of finding logs is still quite effective and actually very good in terms of understanding logarithms, calculators provide more accuracy and efficiency.
And in math, accuracy and efficiency is everything.
And so concludes the last little bits of math you’ll be getting out of me this year. I’d say it’s been a really
long and endless interesting run, wouldn’t you?
You know you would.
More math next year, definitely. Have a good summer and stay awesome, everyone!