Comparing exponential and logarithmic functions Video Lecture - Engineering Mathematics

Voiceover:What I want to do in this video
is graph up a classic exponential function
and then graph a related
logarithmic function
and see how the two are related visually.
The two things I'm going to graph
are y is equal to two to the x power
and y is equal to the log base two of x.
I encourage you to pause the video,
make a table for each of them
and try to graph them
on the same graph paper.
See how they are related
and if you see how they're related,
think about why they are related that way.
Let's first start with y
equals two to the x power.
I'm going to make a little table here,
different x values and the
corresponding y values.
x and y, we can start with negative two,
negative one, zero, one, two, three.
In each case y is going to
be two raised to these power.
Two to the negative two power
is going to be 1/4.
Two to the negative one power is 1/2.
Two to the zero power is one.
Two to the first power is two.
Two to the second power is four.
Two to the third power is eight.
Let's graph that.
Two to the third power is eight.
Two to the second power is four.
Two to the first power is two.
Two to the zeroth power is one.
Two to the negative one power is 1/2.
Two to the negative two power is 1/4.
Even the two to the negative third power
is going to be 1/8,
so it's going to look something like this.
The graph is going to
look something like this
right over here.
It's kind of your classic,
sometimes this will be called
your exponential hockey stick
because it kind of looks
like a hockey stick
where it just kind of starts kind of slow
and just oohh bam, shoots straight up.
Notice as we go to the left
as x becomes more and
more and more negative
our value approaches zero
but never quite gets there.
If we have two to the
negative one millionth power
it's going to be a
very, very small number,
very, very close to zero
but it's not going to be quite zero.
We're going to have a
horizontal asymptote at y
is equal to zero
or the x-axis is a horizontal asymptote.
Fair enough.
Now let's graph y is equal
to log base two of x.
Before I graph that, let's
just think about another way
of representing it.
This literally says, for any x,
what power, what exponent
y if I raise two to that
would give me x.
This is an equivalent statement
as saying two to the
y power is equal to x.
If you notice, what we've done here
between these two things
you're essentially just
switching the x's and the y's.
Here's two to the x power is equal to y.
Here's two to the y power is equal to x.
Really this and this you've
swapped the x's and the y's.
What we will see is
that we can essentially
swap these two columns.
x and y,
so let me just do 1/4, 1/2, one, two,
four, and eight.
Here now we're saying if x is 1/4,
what power do we have to raise two to,
to get to 1/4.
We have to raise it to
the negative two power.
Two to the negative one
power is equal to 1/2.
Two to the zero power is equal to one.
Two to the first power is equal to two.
Two to the second power is equal to four.
Two to the third power is equal to eight.
Notice all we did,
as we essentially swapped
these two columns,
so let's graph this.
When x is equal to 1/4, y
is equal to negative two.
When x is 1/2, y is equal to negative one.
When x is one, y is zero.
When x is two, y is one.
When x is four, y is two.
When x is eight, y is three.
It's going to look like this.
Notice, I think you
might already be seeing
a pattern right over here.
These two graphs are essentially
the reflections of each other.
What would you have to reflect about
to get these two?
Well you'd have to reflect
about y is equal to x.
If you swap the x's and the y's,
another way to think about,
if you swap the axis you
would get the other graph.
It's essentially what we're doing.
Notice it's symmetric about that line
and that's because these are essentially
the inverse functions of each other.
One way to think about it is we swapped
the x's and y's.
Just as this, as x becomes more and more
and more and more negative
you see y approaching zero.
Here you see is y is becoming
more and more negative
as x is approaching zero,
or you could say as x approaches zero
y becomes more and more and more negative.
The whole point of this
is just to give you an appreciation
for the relationship between
an exponential function
and a logarithmic function.
They're essentially
inverses of each other.
You see that in the graphs,
they're reflections of each other
about the line y is equal to x.