(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)

## No Comments