(upbeat music)
- In this video, I wanna
talk about applications
of logarithms.
There are several common equations
that natively contain logarithms,
almost all of them deal with the concept
of comparing quantities
that change a great deal in magnitude.
The first one is pH, pH measures
the acidity of a solution,
and is given by the equation pH equals
and that's lowercase p
capital H equals negative log
and then a bracket.
The reason it's a bracket
is because in chemistry,
we use brackets to denote concentration.
And so it would normally be a parenthesis,
a bracket is just a form of parenthesis.
It's the way this equation is written.
Anyways, let me start over again.
pH equals negative log left bracket,
capital H plus in the
exponent right bracket.
In layman's terms, pH is the negative log
of the hydrogen ion concentration.
So H plus in brackets is the
hydrogen ion concentration
in a solution which is
measured in moles per liter.
A pH of seven is neutral.
I thought I'd give you a
sense for what these numbers
actually mean in terms of scale
in a little table of values.
We have four columns,
in the first two columns
I've written the hydrogen
ion concentration,
that's left bracket H plus, right bracket
in moles per liter.
I've written it in two ways.
The first way is as a
decimal and the second way
in scientific notation.
Then in the third column, we have pH
like what is the actual pH value
and what does that mean
in words? So here we go.
On the first line, if the
hydrogen concentration
is .001 moles per liter or one times 10
to the negative third
in scientific notation,
the pH is three and that's
a very acidic solution,
something like stomach acid, maybe.
If the hydrogen concentration is .00001
or one times 10 to the
negative fifth moles per liter,
then the pH is five, which is acidic,
but not as acidic as stomach acid.
In the third line, we have
a hydrogen concentration
of .0000001 moles per liter,
or one times 10 to the negative seventh
in scientific notation.
This is a pH of seven and this is neutral.
This is like the pH of water.
On the next line we have .000000001
or one times 10 to the negative
ninth, that's a pH of nine.
It's a somewhat alkaline
solution, but not super alkaline.
Finally, we have a
hydrogen ion concentration
of .00000000001 moles per liter,
or one times 10 to the negative 11th.
That's a pH of 11 and
that's very alkaline.
Now you can imagine that
if we just had a scale
that went from 0.001 to .000000000001,
this would be very hard to graph
and it would be very hard to talk about.
So the negative log actually transforms
those very, very tiny
numbers to a numerical pH
that is easy for us to talk about.
Notice that written in
scientific notation,
it's much easier to see the connection
between the power on the 10 and the pH.
The Richter Scale, which
is used in earthquakes
is also measured using logarithms.
The Richter Scale was designed
with two principles in mind.
First, each change of 1.0 on the scale
should be 10 times greater
in earthquake force
than the previous value of the scale.
So a magnitude five earthquake
should be 10 times greater
than a magnitude four earthquake.
Second, an earthquake
with a magnitude of zero
should actually be
undetectable for humans.
And it turns out that both
a magnitude of zero and one
are pretty much not felt by humans.
The formula for the
magnitude of an earthquake
using the Richter Scale is
capital M equals log of A.
So log left paren A, right paren.
A is a capital A where we
measure the maximum amplified
ground motion, capital A in microns.
Now note a micron is a
millionth of a meter.
Again I've made a little table of values
to give you some sense of what
this scale actually means.
In the first two columns, we
have the movement of the earth
in microns, either as a whole number
or in scientific notation.
The third column is the
magnitude of the earthquake
and the fourth column is
the description of it.
So for example, a magnitude
one or one times 10 to the zero
is a magnitude of zero
on the Richter Scale
and it's not felt.
A movement of 10 microns or
one times 10 to the first
is a magnitude one earthquake
and is also not really felt by humans.
A movement of the earth of 100 microns,
which is one times 10 to the second
is a magnitude two
earthquake and is minor.
A movement of 1000 microns
or one times 10 to the third
is a magnitude three
earthquake, also minor.
10,000 microns, or one
times 10 to the fourth
is a magnitude four, which
is a small earthquake.
Also a small earthquake is 100,000
or one times 10 to the
fifth, a magnitude five.
A moderate earthquake is from
1 million microns of movement
or one times 10 to the six,
that's a magnitude six.
A strong earthquake is 10
million microns of movement,
one times 10 to the seventh
and that's a magnitude seven earthquake.
And finally, a great
earthquake is 1 billion microns
of movement, one times 10 to the ninth
and that's a magnitude nine.
Not sure why I skipped eight
there, but we'll go on.
Graphing either the pH
scale or the Richter Scale
in Desmos is very challenging
because of the widely
varying input values.
Let's just graph the earthquake function
and see how well we do.
So we're gonna graph M equals
log of capital A in Desmos.
If we graph this in a
standard viewing window,
we see a logarithmic function.
It has a vertical asymptote on the y-axis
and it's gradually growing.
But in a standard viewing window
and let's just look from zero
to 10 for the Richter Scale
on the y-axis and maybe
like from zero to 15
on the x-axis, if that's all we look at,
we actually can't see any
values of the Richter Scale
above 1.2 on this graph.
So 1.2 is right around
15 microns of movement.
So we're not getting a very
good view of the scale.
Now, Utah experienced a
magnitude 3.7 earthquake
in February of 2019, that's where I live
and look how far we have
to zoom on the x-axis
to see a magnitude 3.7 earthquake.
We have to go all the way to 5012
to see that 3.7 magnitude earthquake.
We haven't altered the y-axis at all,
but that's a huge zoom on
the x-axis and now the values
from like zero to 15 are
completely washed out
for microns of movement.
This is the problem with these scales
that involve magnitudes of 10
and this is why we use logs
to take those scales down
in size a little bit.
Now I have a little tip for you in Desmos
to make this a little bit easier.
Let's go over to Desmos and I'll show you.
I have the magnitude of
the earthquake graphed,
M equals log of A, and
I've got the Richter Scale
on my y-axis.
I'd like to zoom way way in on the x-axis
without affecting the y-axis at all.
And you could do this with a keyboard.
You can either do it of course,
in that wrench menu we always talk about,
but another way to do it is
to press down the Shift button
and hover over the axis.
As soon as you press down Shift
and you get near that axis,
you'll see a left right arrow appear
and then you can click
and drag it to change
that one axis without affecting the other.
So again, we click and drag.
There we go, so you can see,
we can zoom in quite a ways
without affecting the y-axis.
All right, I have a couple
of questions for you to try.
Let's see if you have the
hang of these two applications
around pH and Richter Scale.
Pause the video and give these a try.
Okay we're back.
Let's start by calculating the pH of blood
if the concentration of
hydrogen ions is four times 10
to the negative eighth moles per liter.
We know the pH equation
is lower case p capital H
equals negative log left bracket,
capital H plus right bracket.
In this case, we know the
concentration of hydrogen ions.
So let's rewrite this,
lower case p capital H
equals negative log of left
parenthesis four times 10
to negative eighth, right parenthesis.
Now you can put this into Desmos,
negative log four times 10,
and then we'll use the eight
at the B key, negative eight.
This gives us a value of 7.3979
or I'm gonna round to 7.4.
The pH of blood is
around 7.4. Next problem.
On November 30th, 2018, there
was a magnitude 7.1 earthquake
centered about 10 miles
north of Anchorage, Alaska.
Let's find the maximum amplified
ground motion in microns.
So in this case, we have the
capital M value, the magnitude,
and we want to find the capital A value.
The formula is capital
M equals log left paren,
capital A, right paren.
So we have 7.1 equals log of capital A.
Now how do we solve a
log equation like this?
Well, the base on the
log is a base 10, right?
So we want to raise both
sides into the exponent of 10.
We would write 10 to the
something on the left
and 10 to the something on the right.
That something is a set of
open and close parentheses
with space in between.
So on the left, that's 10
to the 7.1 and on the right,
that's 10 to the log of A.
Well, now I have a base,
that's 10 and a log
that has a base 10 so on the
right side of the equation,
that should simplify to just be capital A.
Now we have 10 to the 7.1
power equals capital A
and we just need to calculate
10 to the 7.1 power,
which is 12,589,254 microns.
That is the maximum
amplified ground motion.
For the third problem, we're
being given that the pH
of stomach acid is between 1.5 and 3.5.
We wanna find the range of
the concentration of hydrogen
that corresponds to this
pH range of 1.5 to 3.5.
Let's just start by
rewriting the pH equation.
Lowercase p capital H equals
negative log left bracket,
capital H plus, right bracket.
In this case, we have the pH,
we wanna find the
concentration of hydrogen ions.
Let's start with a pH of 1.5.
1.5 equals negative log,
left bracket, capital H plus,
right bracket.
Now, usually when we do
the 10 to the on both sides
to solve a log equation,
we only have a log,
we don't have something
in front of the log
like this negative.
That's simple enough to fix,
let's just divide both sides
by negative one, that moves
the negative to the left side.
1.5 divided by negative
one gives us negative 1.5
equals positive log left
bracket H plus right bracket.
Okay, now we know that
this is a log base 10,
we're going to use 10 to
the something on the left
and 10 to the something on
the right as our inverse.
So 10 to the negative 1.5 on
the left and 10 to the log
of left bracket, H plus
right bracket on the right.
On the right, we have 10
to the, so a base of 10
and log, which is also a base of 10.
So that should simplify it
to just be left bracket,
capital H plus right bracket
on the right hand side.
And on the left hand side,
we'll have 10 to the negative 1.5.
Well, I should be able to
calculate 10 to the negative 1.5.
That is 0.0316 for the
concentration of hydrogen ions
and that's in moles,
mol/L, so moles per liter.
So now we've found the
hydrogen ion concentration
that corresponds to 1.5.
We still need to find the
hydrogen ion concentration
that corresponds to 3.5.
Well, if you just follow
this formula through,
let's look at where that 1.5 shows up.
It shows up as 1.5 equals
negative log of H plus,
and then negative 1.5 equals
positive log of H plus
and then 10 to the negative
1.5 on the left hand side.
And that's what we end up
evaluating for our answer.
So I think we can just
do 10 to the negative 3.5
to find the other side of the range
and 10 to the negative 3.5
is 0.0003 moles per liter.
So the range of
concentration of hydrogen ion
for stomach acid would be between,
I'm gonna write this as an inequality.
So I'm gonna put bracket capital
H plus, close the bracket
in the middle, and I'm gonna
put a less than or equal to
on each side.
The bottom number is actually
the 0.0003 moles per liter
and the top number is
0.0316 moles per liter.
The concentration of
hydrogen ions has to be
between .0003 moles per liter
and .0316 moles per liter.
