Imagine we’re on a trading ship setting out from Europe bound for Asia sometime in
the seventeenth century.
The boundaries of the known world are expanding faster than ever and it’s an exciting time
to be alive.
Well, if we can stay alive.
Sea travel is difficult, dangerous, and many ships never make it home.
Luckily our ship is carrying the latest invention that improves our odds.
It’s not a new medicine or a new kind of sail, or anything that looks like new technology.
It’s a little book of logarithms, and it’s going to save the lives of everyone on board.
Is it just me, or do there seem to be a lot of watery graves in this algebra class?
Ahoy! I’m James Tanton, and this is Study Hall Algebra presented by Arizona State University
and Crash Course.
When seventeenth-century explorers used trigonometry -- which is all about the relationships between
triangles -- and the stars to navigate at sea,
they needed to do many long and complicated multiplications.
And they needed to do them fast so they didn’t wreck their ships on the rocks or get lost
and die of scurvy.
Trade and navigation were big business, so mathematicians around the globe were trying
to find ways to speed up the calculations,
like by turning multiplication into addition. But this transfiguration proved harder than
it might sound -- and it sounds pretty hard!
Some came close, but it took until 1614 for a mathematician named John Napier to publish
this revolutionary work.
And so the logarithm was born.
Napier was actually thinking about ratios and particles moving away from each other,
and created the word logarithm
by combining the Greek words for “ratio” and “number.”
It took over a century for the famous mathematician Leonard Euler to figure out that
Napier was really describing the behavior of exponents or powers –
which is how we think about logarithms today.
A logarithm is a power that some fixed number, called the base, must be raised to in order
to get a given result.
For example, log base ten of one-hundred thousand is the power of ten that gives 100,000.
And log base 2 of 8 is the power of two that gives the answer 8.
The base of a logarithm can be most any number, but there are a few that we rely on more than others.
For instance, the log key on your calculator is actually base-10, which is called the common
logarithm
and used most often since we like thinking about powers of 10.
If you see a logarithm and there’s no base specified, that means it’s a common logarithm.
The Richter scale used to describe the strength of earthquakes is built on common logarithms.
A quake of magnitude 6 has amplitudes in the millions of units.
The Richter scale takes the log of the amplitude of the waves,
which means for each step up the scale the strength increases by a whole power of 10.
Like scientific notation, logarithms help us reframe really big numbers and their growth.
And their creation didn’t just save the lives of explorers,
it opened the door for vast achievements in science.
Today we type everything from basic arithmetic to complicated integrals into a calculator,
but there was a time when scientists could fill pages and pages with calculations and
still not make progress.
Imagine trying to model the solar system with only a pencil and paper. Yeesh.
Logarithms can help anyone with long multiplication.
When we’re multiplying two things together, if we take the log, we can split up that multiplication
and add the log of the first thing to the log of the second thing.
Voila! Multiplication turns into addition.
If we think about logarithms as powers and remember x^a*x^b = x^(a+b),
this isn’t that surprising.
We can do an example to be sure.
Our rule says that log base 2 of 4 times 8 equals log base 2 of 4... plus…. log base 2 of 8.
On one side, we have 4 times 8 is 32, so log base 2 of 32 is asking “what power of 2
equals 32?”
Well, 4 is 2^2, and 8 is 2^3, so 32 is 2 to the 2 + 3 = 5th power.
On the other side, these terms are asking “what power of 2 equals 4?”, and “what
power of 2 equals 8?”
Again we have 2 + 3 = 5.
And if we remember x^a/x^b = x^(a-b), it’s not that surprising to learn we can turn division
into subtraction like this.
So if we have log base 3 of 27/9, we can rewrite it as log base 3 of 27 minus log base 3 of 9.
On the left we’re asking what power of 3 equals 3, which is just 1.
And on the right, we’re asking what power of 3 is 27 minus what power of 3 is 9. 3 - 2
also equals 1.
Trying to accurately model patterns around us has driven much of mathematics and science
forward.
And over the centuries, we’ve learned that lots of complicated phenomena,
like population changes or financial interest rates, can be described with exponential equations.
These equations have a variable as the exponent.
But if we had something like 5^x = 31, we wouldn’t be able to solve that precisely
without logarithms.
We might just guess that x is a little more than 2 but much less than 3, since 5^2 is
25 and 5^3 is 125.
To work with this problem, let’s remember the exponent rule about stacking powers and
use some logarithms.
If we have log(10^2), we’re asking what power we should raise 10 to to get 10^2.
And that's 2.
So in general, if M = 10^b, then log(M) = b. We keep all the same pieces -- we’re just
using a different way to say the same thing.
But then let’s take it one step further and stack some powers.
If we take M^k, that’s just (10^b)^k, which is equal to 10^bk.
So log(M^k) is just log(10^bk) which is equal to b times k, since we’re asking what power
of 10 gives us 10^bk.
Then, we can rewrite b as log(M) to get k*log(M) -- and our rule.
We can take down any power inside a logarithm and multiply it in front.
We’ve worked with common logs here, but the same is true for any base.
So let’s hit 5^x = 31 with a log on both sides to get x log 5 = log 31.
Divide both sides by log 5, and this leaves us with x = (log 31)/(log 5).
That looks pretty convoluted, but it’s precise.
By guessing and checking you might get x is about 2.13, but (log 31/log 5) is the exact form,
the same way square root 2 is the exact form and 1.41 is an approximation.
So in general, we’ve just seen that log_a(a^x) is x and we can use a logarithm to undo an exponent.
Does this work the other way, can exponents undo logarithms?
We want something like a^(log_a(x)) to just equal x.
And thinking about powers, that makes sense if the bases match. log_a(x) is the power
of a that gives the answer x.
So if I plug it in as my power of a, I have to get x!
And to round out our Important Logarithm Properties, notice two things. log_a(a) = 1 for all values of (a),
because this logarithm is really asking “what power do I raise a to in order to get a.”
a to the 1 is a!
And similarly, log_a(1) = 0 for all values of (a), because anything raised to the power
of 0 is 1.
There’s one more special type of logarithm that gets its own name: the natural log.
That’s a logarithm with base e – the irrational number approximately equal to 2.718.
Why does a logarithm built on e get a special name?
As we notice more math around us, e keeps popping up, like it’s hard-coded into the universe.
But actually, e shows up so much because we choose to use it. When something like bacterial
growth is doubling and is described by an exponential equation like 2^x, scientists discovered
that its growth rate was equal to 0.693 times the population size: an awkward number.
Growth based on tripling, or 3^x, had a growth rate of 1.099 times the population size: another
awkward number.
Logically, there must be something between 2 and 3 where the growth rate of the exponential
equation is just 1 times the population size.
After much searching, it turns out e is the number that makes the mathematics of exponential
growth easiest.
To get a sense of just how prevalent this number e is, let’s look at the world through
… X, Y, and e-ray Vision!
Through the Math Goggles, we can see e is absolutely everywhere.
The changes in height of sunflowers growing in a garden,
the amount of heat lost by a pie cooling on a windowsill,
the amount of interest accumulating in a savings account.
And everything that contains radioactive isotopes, like nuclear reactors,
amoebas, mathematicians, and stars… and dead stuff.
In fact, measuring the amount of radioactive carbon-14 isotopes is called carbon dating.
And it’s used to establish the age of very old artifacts.
In any living organism, the ratio of carbon-14 to ordinary carbon is about constant.
But once something’s dead and no longer taking in any carbon, the amount of carbon-14
decreases exponentially
-- and we don’t just mean “really fast”.
It actually decreases proportional to its starting point.
The model for this process typically looks like this.
The amount of carbon at some time t is equal to the initial amount of carbon times e raised
to the rate of decay multiplied by time t.
Let’s say we’ve found an old-looking piece of intricately carved wood in a cave,
and we find its carbon-14 ratio is about 45% of that in living wood.
So our current carbon is 45/100, or 0.45, but initial carbon was 100/100 at the time
the wood was cut, or just 1.
And scientists have found that the rate of decay -- or decrease -- of carbon-14 is approximately
−0.0001216.
Plugging everything in, our equation looks like 0.45 = e^(–0.0001216*t).
If we get time out of the exponent and solve, we’d have a pretty solid guess on how old
this artifact is.
Hit it with a natural log!
The log and e cancel out since log_a(a) is 1 for any number a.
So solving for time t gives us ln(0.45)/(−0.0001216).
If we calculate that, we get approximately … 6,566.7 years.
Whoa! We need to talk to Indiana Jones about this thing.
Logarithms sound mysterious but are incredibly powerful tools -- and they always have been!
But they definitely look confusing. So if you’re stuck, here’s what you can do.
If you want to pull a variable down to earth and take it out of an exponent, hit it with
a logarithm.
But if you want to get rid of a logarithm, exponentiate using the same base.
And If you’re modelling growth or decay, you’re probably going to use e.
If you’re describing something in orders of magnitude, you’re probably going to use
log base-10.
If there’s no logarithm in your equation before you introduce one, you can use any
base you like. So we’ve expanded our range by quite a lot
since our first video.
We’ve gone from counting to finding new types of numbers to turning multiplication
into addition and somehow not breaking everything along the way.
But there’s always more math to learn.
Next time we’ll look at polynomials and imaginary numbers – the next puzzle piece
in this story of algebra. Until then, cheers!
Thanks for watching Study Hall Algebra, which is produced by Arizona State University and
the Crash Course team at Complexly.
If you liked this video and want to keep learning with us here in Study Hall, be sure to subscribe.
You can learn more about ASU and the videos produced by Crash Course
in the links in the description.
See you next time!