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Why Do We Learn Logarithms?

Date: 02/28/2005 at 11:31:45
From: Gary
Subject: What is the point of logarithms?????

I am an 8th grader currently studying logarithms in school and I was 
wondering; what is the point of them.  Did one guy say one day, "Hey, 
let's torture children."  Will we ever use them when we grow up, and 
how are they used in real life?  Why do we learn them at all?

I don't understand why we learn these and they confuse me about 
mathematics.  What is the point of finding the logarithms of 5 in base 
7?  Can you please help me out, because I am starting to lose interest 
in math.

Date: 02/28/2005 at 11:58:41
From: Doctor Jerry
Subject: Re: What is the point of logarithms?????

Hello Gary,

Logarithms were invented way back, in the 1500s or 1600s.  At that 
time, calculators didn't exist.  To do multiplications, divisions, and
root extraction with numbers having five or more digits required a lot
of time and work.  Logarithms reduced the needed amount of work by a
large amount, way more than half I'd guess.

Now that calculators and computers are common, logarithms are still
very, very useful, but in a totally different way.  They are very
closely related to exponential functions.  For example, you probably
know by now that  y = a^x is equivalent to log_a(y) = x, where
log_a(y) is the base a logarithm of y.  The exponential function y=a^x
is one of the most important functions in mathematics, physics, and
engineering.  Radioactive decay, bacterial growth, population growth,
continuous interest,..., all involve exponential functions.  Because
of the relation  that y = a^x is equivalent to log_a(y) = x, logs are
equally important.

With regard to different bases, base 10 is used by chemists in their
measurements of pH, the acidity of a liquid; base 2 is used in 
information theory and computers (used in transmitting information and 
measuring the errors made and how to correct those errors); base e,
where e=2.718281828..., is used in calculus and is probably the most
important base.  Base 5 or base 7 have no real importance, other than
providing practice in working with logarithms.

You, personally, may never use logarithms; but the guy next to you may
become a scientist or an engineer.  Maybe you will.  It's hard to tell
in 8th grade.

If you have any questions about my comments, please write back.

- Doctor Jerry, The Math Forum 

Date: 02/28/2005 at 12:18:20
From: Doctor Ian
Subject: Re: What is the point of logarithms?????

Hi Gary,

Historically, logarithms are interesting because they allow you to use
addition to do multiplications.  For example, if you want to multiply
365.49 by 1474.3, you can do this:

         x = 365.49 * 1474.3
  log  (x) = log  (365.49 * 1474.3)
     10         10

  log  (x) = log  (365.49) + log  (1474.3)
     10         10              10

  log  (x) = log  (10^2 * 3.6549) + log  (10^3 * 1.4743)
     10         10                     10

  log  (x) = 2 + log  (3.6549) + 3 + log  (1.4743)
     10             10                  10

  log  (x) = log  (10^5) + log  (3.6549) + log  (1.4743)
     10         10    10      10              10

Now, that's cool, because you can look up the logarithms of the
numbers 3.6549 and 1.4743 from a table; add them; and then look up the
inverse logarithm of the result.  Then multiply by 10^5 (by shifting
the decimal place) to get the final product. 

This isn't such a big deal now, when cereal boxes give away 
calculators that can do it; but it's what made slide rules possible,
and before that, it made a lot of calculations easy that would have
been much more difficult.   

Why should you care about them now?  There are a few situations where
logarithms are in common use. In chemistry, pH (a very basic concept)
is defined in terms of logarithms.  In physics, they are used for
calculations involving radioactive decay.  In biology, they are used
for modeling population growth.

Think about what makes division useful.  There's nothing you can do
with division that you can't do more clumsily using multiplication in
conjunction with guess-and-check.  When you just want to multiply things, 

   16 * 48 = ?

that's easy.  When you only have one product and a result, 

   17 * ? = 592

division lets you invert the multiplication:

   17 * ? = 592   ->  592 / 17 = ?

Logarithms do the same sort of thing for exponents.  Sometimes you
just want to raise something to an exponent, and that's easy:

   12^4.9 = ?

But sometimes you have the base and the result, and you need to find
the exponent:

   13^? = 6.3

You _could_ figure this out with guess-and-check; but logarithms let
you get the result directly:

   13^? = 6.3   ->   log  (13^?) = log  (6.3)
                        13            13

                               ? = log  (6.3)

But even if you graduate and forget how to do actual calculations with
logarithms, it's still worth knowing _about_ them, because they are
often used in various scales of measurement.  For example, the Decibel
scale (which measures the intensity of sound) and the Richter scale
(which measures the intensity of earthquakes) are based on logarithms.  

Because these are logarithmic scales, the changes at one end of the
scale are much, much more signifcant than the changes at the other
end.  For example, change from 1 to 2 on either scale is much, much
less significant than a change from 9 to 10.  If you don't understand
that, you won't be able to make sense of a lot of information that
could directly impact your health and safety. 

>I don't understand why we learn these and they confuse me about 
>mathematics. What is the point of finding the logarithms of 5 in base 
>7? Can you please help me out, because I am starting to lose interest 
>in math.

There are a few different ways to look at it.  One is that you're 
being taught this stuff because your teachers know how to teach it. 
That's the cynical view, but there's probably a lot of truth to it. 
If you take this view, you have a couple of choices:  learn it and get
good grades, or refuse to learn it and get lousy grades.  

Another way to look at it is that when you're learning math, you're
supposed to be learning a certain way of thinking, which is largely
about how ideas can lead to notations, and how those notations can
lead to new ideas, which lead to new notations, and so on. 

For example, if you didn't know anything about logarithms, you could
invent them if you understood the concept of an inverse.  That is, you
could ask the question:  "Some operations have inverses, e.g.,
subtraction inverts addition, and division inverts multiplication.  I
wonder if there's anything like that for exponents?"  And that would
lead you very naturally to figure out what logarithms are, and how
they work.  

This is a very empowering way of looking at the subject, for a lot of
reasons.  For one, it sharply reduces the amount of memorization you
have to do.  If you understand where ideas come from, there's nothing 
to memorize, because you can figure out all the rules from first
principles.  For another, being able to see each new idea in the 
context of a larger overall picture makes the whole thing seem...
well, interesting.  And even fun.  It starts to feel like something
you could do on your own, instead of something you have to wait around
for others to teach you.  If you ever get to that point, your
difficulties with math are over, for good. 

A third way to look at it is this:  You just never know what's going
to turn out to be useful:

  Math is Power? 

That goes for logarithms, too. 

I hope this helps!  Write back if you'd like to talk more about this,
or anything else. 

- Doctor Ian, The Math Forum 
Associated Topics:
High School Logs
Middle School Logarithms

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