What is a logarithm?
Take the equation 10x = 1,000,000. Solving for x
in this equation is the same as solving for x in the following equation: log10(1,000,000)
= x. This can be stated as, “x is the logarithm of 1,000,000 to base 10.”
Logarithms changed the world because they allowed for time
consuming multiplication and division of large numbers to be reduced to table
look-ups and addition or subtraction. This is because the following is true:
logb(xy) = logb(x) + logb(y)
and
logb(x/y) = logb(x) - logb(y)
Now imagine you are an early 17th century
engineer spending a majority of your time doing tedious multiplication and
division problems. Along comes John Napier with a bunch of look-up tables that
give the log value for numbers to the base (1-1/10^7)^(10^7) which we can call
b.
You have these two large numbers x and y that you need to
multiply. Thanks to John’s laborious pre-work, you can now look up logb(x)
and logb(y) in seconds. With a much faster to calculate addition
problem and a reverse of the previous look-up, you now have xy.
It is hard to overstate the importance of this innovation.
Just consider the human accomplishments that utilized a version of this
tool from Napier’s first tables in 1614 to the retirement of the slide rule in
the 1960’s; these accomplishments include: Johannes Kepler’s third law of planetary motion,
commercial airliners and everything in-between.
I went off on a tangent about the history of logarithms when
this article is supposed to be focused on natural log. Speaking of tangents, a tangent
line is just what makes the natural log so natural. If you think about the
graph of the two columns in your new look-up tables, that is y = logb(x),
it must pass through the point (1,0) because logb(1) = 0 just as b0=1
for any base b. The slope of the tangent line to y = logb(x) at
(1,0) depends on the base b.
What base b do you think has a tangent line at (1,0) with a
slope of 1?
Why the answer is e, naturally.
Thank you Leonard Euler.
For the sake of brevity, I’ve stated a lot without proof and
simplified the history a bit. Here are some great links to learn more:
1 comment:
Heh. "Speaking of tangents..." ...goes off on tangent about tangents.
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