Welcome to the logarithm
presentation. Let me write down the word
logarithm just because it is another strange and unusual
word like hypotenuse and it's good to at least, see it once. Let me get the pen
tool working. Logarithm. This is one of my most
misspelled words. I went to MIT and actually one
of the a cappella groups there, they were called
the Logarhythms. Like rhythm, like music. But anyway, I'm digressing. So what is a logarithm? Well, the easiest way to
explain what a logarithm is is to have first-- I guess it's
just to say it's the inverse of taking the exponent
of something. Let me explain. If I said that 2 to the third
power-- well, we know that from the exponent modules. 2 the third power, well
that's equal to 8. And once again, this
is a 2, it's not a z. 2 to the third power is 8, so
it actually turns out that log-- and log is short
for the word logarithm. Log base 2 of eight
is equal to 3. I think when you look at that
you're trying to say oh, that's trying to make
a little bit of sense. What this says, if I were to
ask you what log base 2 of 8 is, this says 2 to the
what power is equal to 8? So the answer to a logarithm--
you can say the answer to this logarithm expression, or if you
evaluate this logarithm expression, you should get a
number that is really the exponent that you would have
to raised 2 to to get 8. And once again, that's 3. Let's do a couple more examples
and I think you might get it. If I were to say log--
what happened to my pen? log base 4 of 64 is equal to x. Another way of rewriting this
exact equation is to say 4 to the x power is equal to 64. Or another way to think
about it, 4 to what power is equal to 64? Well, we know that 4 to
the third power is 64. So we know that in this
case, this equals 3. So log base 4 of
64 is equal to 3. Let me do a bunch of more
examples and I think the more examples you see, it'll
start to make some sense. Logarithms are a simple idea,
but I think they can get confusing because they're the
inverse of exponentiation, which is sometimes itself,
a confusing concept. So what is log base 10 of
let's say, 1,000,000. Put some commas
here to make sure. So this equals question mark. Well, all we have to ask
ourselves is 10 to what power is equal to 1,000,000. And 10 to any power is actually
equal to 1 followed by the power of-- if you say 10 of
the fifth power, that's equal to 1 followed by five 0's. So if we have 1 followed by six
0's this is the same thing as 10 to the sixth power. So 10 to the sixth power
is equal to 1,000,000. So since 10 to the sixth power
is equal to 1,000,000 log base 10 of 1,000,000 is equal to 6. Just remember, this 6 is an
exponent that we raise 10 to to get the 1,000,000. I know I'm saying this in a
hundred different ways and hopefully, one or two of these
million different ways that I'm explaining it actually
will make sense. Let's do some more. Actually, I'll do even a
slightly confusing one. log base 1/2 of 1/8. Let's say that that equals x. So let's just remind
ourselves, that's just like saying 1/2-- whoops. 1/2. That's supposed to
be parentheses. To the x power is equal to 1/8. Well, we know that 1/2 to the
third power is equal to 1/8. So log base 1/2 of
1/8 is equal to 3. Let me do a bunch
of more problems. Actually, let me mix
it up a little bit. Let's say that log base
x of 27 is equal to 3. What's x? Well, just like what we did
before, this says that x to the third power is equal to 27. Or x is equal to the
cubed root of 27. And all that means is that
there's some number times itself three times
that equals 27. And I think at this point
you know that that number would be 3. x equals 3. So we could write log base
3 of 27 is equal to 3. Let me think of
another example. I'm only doing relatively small
numbers because I don't have a calculator with me and I
have to do them in my head. So what is log-- let
me think about this. What is log base 100 of 1? This is a trick problem. So once again, let's just
say that this is equal to question mark. So remember this is log
base 100 hundred of 1. So this says 100 to the
question mark power is equal to 1. Well, what do we have to
raise-- if we have any number and we raise it to what
power, when do we get 1? Well, if you remember from the
exponent rules, or actually not the exponent rules, from the
exponent modules, anything to the 0-th power is equal to 1. So we could say 100 to
the 0 power equals 1. So we could say log base 100
hundred of 1 is equal to 0 because 100 to the 0-th
power is equal to 1. Let me ask another question. What if I were to ask you
log, let's say base 2 of 0? So what is that equal to? Well, what I'm asking you,
I'm saying 2-- let's say that equals x. 2 to some power x
is equal to 0. So what is x? Well, is there anything
that I can raise 2 to the power of to get 0? No. So this is undefined. Undefined or no solution. There's no number that
I can raise 2 to the power of and get 0. Similarly if I were to
ask you log base 3 of let's say, negative 1. And we're assuming we're
dealing with the real numbers, which are most of the numbers
that I think at this point you have dealt with. There's nothing I can raise
three 3 to the power of to get a negative number,
so this is undefined. So as long as you have a
positive base here, this number, in order to be defined,
has to be greater than-- well, it has to be greater
than or equal-- no. It has to be greater than 0. Not equal to. It cannot be 0 and it
cannot be negative. Let's do a couple
more problems. I think I have another
minute and a half. You're already prepared to do
the level 1 logarithms module, but let's do a couple of more. What is log base 8-- I'm
going to do a slightly tricky one-- of 1/64. Interesting. We know that log base 8 of
64 would equal 2, right? Because 8 squared
is equal to 64. But 8 to what power
equals 1/64? Well, we learned from the
negative exponent module that that is equal to negative 2. If you remember, 8 to the
negative 2 power is the same thing as 1/8 to the 2 power. 8 squared, which
is equal to 1/64. Interesting. I'll leave this for
you to think about. When you take the inverse of
whatever you're taking the logarithm of, it turns
the answer negative. And we'll do a lot more
logarithm problems and explore a lot more of the properties of
logarithms in future modules. But I think you're ready at
this point to do the level 1 logarithm set of exercises. See you in the next module.