The goal of this video is to introduce
ourselves to logarithms.
Well, logarithms basically are the
combination of the last two things we've
done. A logarithm is the inverse of an
exponential. So we dealt with inverse
functions back in 4.1, well 2.6, and then
we played with exponential functions in
4.1 and 4.2. Now we're looking at the
inverse of an exponential. And so we're
going to think about it first by domain
and range, and let's compare them.
Exponential functions have this, so
logarithms will have an equivalent over
here. Let's make that line the whole way
across. Exponential had a domain - now this
is non-transformed exponential, so it's b
to the x - had a domain of negative
infinity to infinity, which means
logarithms will have a range of negative
infinity to infinity. Remember, the domain
and the range, between a function and its
inverse,
just switch places. Which means the
exponential function's range of 0 to
infinity gives us a domain of the
logarithm, 0 to infinity. Now again
remember, this is a non transformed
logarithm. We will have to play with
domains a bit more intensely later on.
Exponential functions always contained
the point (0, 1). Because
if I take b and raise it to the zeroth
power, I get one. So I should have said
equals y. Well, then the logarithm. How did we switch from a function to its inverse?
We swapped x and y. So it contains the
point (1, 0). By the same argument I'd say
the exponential contains the point (1, b),
because b raised to the first power is b,
it doesn't matter what b is. So the
logarithm would contain the point (b, 1).
Swap x and y. The exponential had a
horizontal asymptote on the x-axis, which
means it's equation was y equals 0. Well
if we swap x and y, we're going to get a
y-axis asymptote, x equals 0. That means
it is a vertical asymptote. And then in
general, we remember inverses... If we have f of x equals y, well that means f
inverse of y equals x. Remember, it just
undoes what happened before. The inverse
simply undoes what the function did. And
so specifically, that means, well our original
function was an exponential, so we had
something like b to the x equals y. Well
I'm gonna change it a little bit. That's
equivalent - this symbol is equivalent to -
so that's equivalent to saying log base
b of y equals x. And now notice I read
that "log base b of y equals x," and that
is important.
Saying log b to the y, or log of b to the y
is actually something completely
different. We will look at that a little
bit later on. This is log base b of y
equals x. Ok, so let's play with that a
bit. For example, if I have 5 to the 4th
equals 625 - go ahead and double check
that that's correct, you can pause the
video - well if I have that, according to
what we just said this is equivalent to
writing log base, what's my base? In the
exponential, the base was the thing being
raised to a power. So my base was 5, which means my base of the logarithm is also 5.
Notice, this is written as a subscript.
The base of a logarithm is a subscript.
And then x and y switched places. Here, y was on the opposite side of the equals sign
from my function; here, y is the input of
my function. So what was on the opposite
side?
625. That means we want log base 5 of 625,
and that's equal to our previous input,
which was 4. So all logarithms do is undo
exponentials.
For example, let's evaluate log base 4 of
64. How am I going to do that? Honestly, I
don't think about logarithms by
themselves - I always think about them as
exponentials. So what I'm going to do... So I'm going to think about it as if I
wrote log base 4 of 64 equals, we'll say
x, and then I'm going to rewrite this as
an exponential. So what's my base? Right
here it's 4. And now these trade places,
so instead of log base 4 of 64 I want 4
to the x, and it's going to equal 64.
These two are equivalent statements.
Saying "log base 4 of 64 equals x" is the
same thing as saying "4 raised to what
power is equal to 64?" and I didn't even
really need that equals x. I needed it in
order to convert equations but I didn't
need it up here. All I had to do was look
at log base 4 of 64 and say, well 4
raised to what power equals 64? And that
power is, of course, 3. So x equals 3, which
means log base 4 of 64 is equal to 3.
Okay, a couple more things to do about logarithms.
First of all, there are two specifically
named logarithms. One is called the
common logarithm. And that looks something like this.
Well I'm going to do it as an example.
We're going to evaluate log of 10,000.
Ooh! What do I do? I don't see a base here. We're missing a base. Well whenever we
see the word "log" with no base, this is
the same thing as saying "log base 10." And
that's what we call the common logarithm. log base 10 is common enough. It's just
like with square root. With a square root
you'd write the radical sign, so you'd
write that radical without any index
here, and we know that means 2. That's our
common radical. Well our common log is
base 10, so we will write it without even
bothering to write the base. And now we
ask 10 raised to what power is 10,000?
Well, with 10 that's easy: we just count
the zeros, and that is four. The other one
is called the natural logarithm, and it
looks like this. I'm going to write it
purely as an equation, y equals natural
log of x, and this is read quote "natural
log of x." Well, the equivalent exponential
form, what's the base? When I see ln, the
base is the natural number e. So that's
the same thing as saying e to the y
equals x.
And the natural logarithm, there's a
button on your calculator that will help
you evaluate that. So let's look at a new
page and now try to graph. How about we
graph h of x equals log base 6 of x. How
am I going to do that? There's a few
things we had from last time. First of
all we know the domain is 0 to infinity,
we know there's a vertical asymptote on
the y-axis. So let's draw my axes over
here. I know I've got a vertical
asymptote on the y axis. I know my domain is 0 to infinity. So I know I don't even
really need that x, negative x-axis there.
We know it goes through the point,
oops - that's exponential. Exponential goes through (0, 1), so the logarithm goes
through (1, 0). Then I claim it goes through (b, 1). Why is that? Well log base 6 of 6, I
would be asking 6 raised to what power
is 6, and that is 1! So, 1 2 3 4 5 6 1. It's
an increasing function still, so it will
look something like this. And we can do
any transformations we want. Just watch
out. If I sketch instead g of x equals
log base 6 of x minus 2, we need to be
careful. Order of operations, tells me I
take log base 6 of x first, and then
subtract 2.
This 2 here is a vertical shift down
2. So that means we would take this
function that we've just graphed, let's
make it blue, and move everything down 2
spots, and we end up with something that
looks like that. If I meant it to be a
horizontal shift, change colors again, I
would need something like f of x equals
log base 6 of, in parentheses, x minus 2.
That will now be a shift to the right.
So just watch out for those parentheses.
It's really easy to mix this one up. One
last little detail. Let's worry about the
domain of a logarithm. Well didn't we say
it's always zero to infinity? That would
be non transformed. Let's look at this
one. I have moved it to the right 2.
That means if I played with it on this
axis, one two one two,
well the asymptote also moves to the
right 2, so it is right here. We have a
graph that looks roughly like that, whose
domain is clearly 2 to infinity.
So how do I play with that? Well what
matters is that the inside of the
logarithm, the input, is positive. So it
doesn't matter if x is positive or not,
just the input. Example: log, we'll do a
common log, 7x plus 35. What's our domain?
Well we need 7x plus 35 to be greater
than zero. So 7x must be greater than
negative 35. So x must be greater than
negative 5. Woah, negative 5. Which means
our domain is negative 5 to infinity.
Thank you for watching.
