 (upbeat classical music) Logarithms. The word itself has a magnitude that has caused
even the most decorated veterans of soil studies to break down and weep. Don’t worry, their fine now. In reality, there’s nothing to fear about logs. We must simply learn to understand them. Soil science uses the logarithmic scale to
predict how much water is available to plants. Let’s start with the scale you’re familiar
with, a linear scale, like the one used to measure distance on a globe. The notches on a linear scale are evenly spaced. On this scale, each notch indicates an increase
of 200 units. Perhaps, the most basic of linear scales is
a scale of one to ten, the go-to rating for measuring anything from a quality of a film
to the attractiveness of another person. Aaaaarow. The scale works just like we expect. The number five sits exactly between zero and ten with the other numbers evenly spaced in between. Linear scales, such as this one, are additive
which means we add a value to increase or decrease on a scale. Logarithmic scales are multiplicative or exponential. That means we multiply a value to increase
it on the scale. So, on a linear scale, we add. Zero plus one is one, plus one is two, plus
one is three. And on a logarithmic scale, we use exponents. Ten to the power of one is ten, ten times
ten is one hundred, ten to the power of three is one thousand. But, why should we use the log scale? Why not just use the linear scale to measure
things? Picture an astronauts house. Now, think about the distance the astronaut
drives from the house to the launchpad. This distance is tiny compared to the distance
that the astronauts rocket travels to the moon. And the distance to the moon is just a small
hop compared to the distance to the sun. If all of these points were placed on a linear
scale, that scale would be very hard to read. Since, the data is extremely spread out for
large values, but really close for small values. A log scale solves this problem by expanding
the small values of data where the points are tight, and compacting all of the large
values where data is spread out. At first glance, the log scale could be confused
with a linear scale. The notches are evenly spaced. But, don’t forget that although there was an
even distance between numbers marked on the scale, they are actually increasing quite
rapidly, well exponentially. You see, this group has a total of nine numbers,
while this has ninety, and this one has nine hundred. Each group has ten times the last. Now, let’s take a look at this space. The numbers in here don’t appear like they
do in the linear scale. Instead, they start far apart and get closer
together as they increase in value. This is how we visually compress all our data. There are many systems that use logarithms:
the galactic scale, the microscopic scale, the decimal scale, photographic f-stops, the
richter scale, and even soil particle size which we’ll look at next. All of these graph information that is measured
exponentially. This graph shows us the particle surface area
in relation to the particle size. The Y Axis measures the specific surface area
of each particle. The X Axis keeps track of particle size, expressed
as a particles diameter or width. As the Y Axis increases by 50,000 with each
notch, the X Axis starts at one ten-thousandth and increases by a factor of ten with each
notch which means we multiple each notch by ten. And we know multiplicative means logarithmic. So, particle size on the X Axis is logarithmic. This set of data is already plotted for us. We can see it on this curve. To find the values at any point on the curve,
we simply draw a line from the curve to each Axis. Let’s find the particle surface area and size
for this point on the curve. We’ll make a line to the Y Axis to find the surface area, that’s about 75,000 meters squared, per gram. Now, we make a line to the X Axis for the particle
size. Remember, this is not a linear scale, it is
logarithmic. So, instead of evenly spaced notches, the
notches compress to the right. We see the compression of two, three, and
four ten-thousandths all the way up to 0.001. It appears that the particle size for our
sample is five ten-thousandths of a millimeter wide. You can find the exact position of these values
by using a calculator. But when estimating, all you need to know
is this pattern and how they are compressed to the one side. And that’s all there is to it. Now you can fearlessly tackle a logarithmic
scale representing any set of data. See, although they may be intimidating at
first, logs are actually helping us, aren’t you, little log. (soft jazz music)