Hi there,
so today I'm going to talk about section 5.1,
which talks about logs and their properties.
Okay, so,
one of our motivating factors
for talking about logarithms is the following:
suppose we wanted to solve something like,
12 to the power of 'x' is equal to 1347, 'kay?
We certainly estimate using trial and error,
just by plugging in different values for 'x',
to at least approximate what 'x' is going to be,
but we have no way to solve exactly for 'x'
and this is something that we're gonna see,
soon enogh, in these upcoming sections
that we're gonna wanna be able to do, 'kay?
So, our goal is to be able
to take an exponential function like this
and be able to solve it, 'kay?
So, that's sort of our motivating factor, 'kay?
So, since our motivation involves exponentials
lets go back to exponentials, 'kay?
Let's just take a very basic exponential function,
say, 'y' equals 'a' to the power of 'x'.
Now, one of the things that we saw
for our general exponential functions
is that our graph looks like this.
'Kay, we have a horizontal asymptote right here,
because, remember, as 'x' gets very small
our function values approach zero, 'kay?
One of the things
that I'm going to mention about this function
is that it's a one-to-one function, 'kay?
Recall, you have a function
if it passes your vertical lines test
In other words, each x-value
goes to exactly one y-value.
The other thing that we've talked about
is one-to-one functions.
We have that horizontal line test
for one-to-one functions, 'kay?
Now, first off,
what does that mean practically speaking?
Well, again, in order to be a function
each x-value has to go to exactly one y-value,
but in order to be one-to-one
each y-value, in other words,
each value in your range
can only be gotten to by one value in your domain.
So, in other words, everybody has a partner
and only one partner, okay?
Now, what was the issue
with a one-to-one function?
Well, remember functions need to be one the one
in order to be able to have an inverse, 'kay?
We saw that if a function wasn't one to one
we could theoretically define an inverse,
but it was not necessarily a function, 'kay?
So, in order for a function
to have an inverse function
he needs to be one to one.
This exponential function is one the one,
therefore, we want to find it's inverse, 'kay?
Well, how to find
the inverse?
Well, recall, you start out with your function
and then our first step
is to switch our 'x' and 'y' values,
in other words, 'x' is equal
to 'a' to the power of 'y', 'kay?
Our next step then,
is now solve for 'y', 'kay?
Well, how do we get 'y' equals here?
Now, some people say, "Well raise both sides
to the one over 'y' power or take the 'y'th root,"
yeah, that would remove it from our right side,
but that would just create a mess on our left side.
In fact, it turns out that there is no way
to do such, 'kay?
So we run into a problem here,
we know an inverse exists,
but we have no way to write it as 'y' equals.
Therefore, what we do
is we say the following, 'kay?
We know the inverse
is equivalent to this statement,
so what we do as we say 'x' equals 'a' to the 'y'
is equivalent to
'y' equals log base 'a' of 'x', 'kay?
This equivalence,
as we saw in that previous section,
is of crucial importance.
Alright, so, again, I know it seems really funky
that we suddenly came up
with 'y' equals log base 'a' of 'x',
but again, remember, we know what inverse exists
and we know that it's equivalent
to this form here, okay?
So, I understand that logs
seem sort of like out from left field,
but I assure you
that while we have this definition here,
using this equivalence,
everything is going to follow.
Now, as I mentioned before,
there are two bases that come up very frequently.
Base 10,
and base 'e'.
They come up so frequently
that instead of writing log base 10
we just write log;
and instead of writing log base 'e',
that's actually called our natural log,
we write natural-'ln' for natural log.
So, a couple quick things to show you is as follows:
when you see 'y' equals log of 'x',
using this equivalence,
that's equivalent to 10 to the 'y' equals 'x'.
Remember, if there's no number written there
by default it's 10,
10 to the power of 'y' equals 'x'.
In a similar manner, if you see 'y' equals log 'x',
the natural log of 'x',
that's equivalent, remember that's a base of 'e',
of 'e' to the 'y' equals 'x', 'kay?
So, just to reinforce this equivalence,
let's just play with a couple examples.
So, during the next two examples,
what we want to do is to rewrite the logs
using exponentials, 'kay?
I mean, practically speaking,
when things are written like this
they don't seem as useful.
So one of the mantras you're gonna have
in this section whenever you see logs
is change to the equivalent exponential form, 'kay?
That sort of thing should ring through your head,
and so, let's just practice with that very quickly.
For example,
if I have log of a thousand equals three, 'kay?
This is a true statement,
but all were actually wanting to do here
is just rewrite this
in equivalent exponential form, 'kay?
So, what do you do,
you don't see a number written here,
we know that that's ten
to the power of three equals a thousand,
and I don't think you'll argue with that statement.
Okay, let's do another one.
What if I have
log of .001 equals negative 3?
Again, we don't have a number written here,
by default its a base of 10,
to the negative 3rd power is .001,
and again, I think you grant me
that that is a true statement.
Alright, now let's go the other way,
we're going to take some exponentials
and were going to rewrite them using logs.
For example,
I'll write 10 to the negative 2nd power.
is .01, 'kay?
Again, we know this to be the case,
remember when you have a negative exponent
you move him down to the denominator,
and, so we have 1/100th,
which is exactly this in decimal form, 'kay?
So how would we write this
in equivalent logarithmic form?
Well, you write the word log,
we see the base of 10
so we don't need to put anything here,
it's not incorrect if you do,
and then, remember, the way you take this
is you say your exponent goes on the other side,
so it's log of. 01
is equal to -2
Okay, one more of these
and then I will make the comment
that I understand that this equivalence
is sometimes a bit difficult at first,
I recommend you have it
written down in front of you
when-as you're doing all of these exercises,
because it-it comes with practice.
Alright, what if I have 10 to the 4th equals 10,000?
Again, this is certainly a true statement,
we have a base of 10 so we'll just write log,
and log of 10,000 equals 4.
Remember, your exponent is your y-value,
that's the thing that's on the other side.
Alright, so,
another comment that I want to make is as follows,
and this is going to go back to the whole idea
that we discussed with, uh,
composition of functions
and inverse functions, 'kay?
One quick thing I want to remind you of,
so I'll say, recall,
when we're dealing with our inverse functions,
what happens, 'kay?
One of the things that we saw is that, um,
our function and it's inverse,
they essentially undo each other.
So, for example,
if you start with a function, I'll call him 'f',
and recall that we write the inverse
as 'f' inverse, 'kay, what happens?
Well, we take a point 'x', we plug him in 'f',
and out comes 'f(x)'.
What happens when we plug this into our inverse?
Remember, by definition,
it takes us back to where we started.
In other words, 'f' inverse of 'f(x)' equals 'x'.
You guys remember that this is exactly
what we did with our composition, 'kay?
Start with the point, plug him in the first function,
plug him in the outer function, and by definition,
if they are inverses,
you get back to where you started.
Recall, that we could do that with a point,
we'll call him 'x',
and we could plug him in the inverse function first,
and then when we plug them into our function,
again, we get back to where we started, 'kay?
Now, let's look at this in the context
of our exponential function
and their corresponding inverse function,
which is a log function.
So, let's first start with this guy,
let's suppose my function is 10 to the power of 'x'
and we know is the inverse is log 'x', 'kay?
I'll even write log base 10 here,
just to reinforce this idea, 'kay?
So, remember,
because they're inverses they undo each other.
So, for example,
if I start out with some point, I'll just call him 'p',
and I plug him in my function 'f'
out comes 10 to the 'p', 'kay?
Now, if I plug him into my function 'g',
I think you'll grant me
that were going to go back to 'p',
but look at what we're doing,
what does it mean
to plug 10 to the 'p' into this function?
It means were taking log base 10
of 10 to the 'p', 'kay?
And because they're inverses
we've got to get back to where we started.
Look at this, we start with 'p',
we plug him into our exponential, 'kay,
and we get that output;
we take that output, plug him into the inverse,
remember, we have to get back
to where we started.
Do you remember
that this is one of the properties?
That when these two numbers match
you just get the exponent, 'kay?
That's where that comes from.
In a similar manner, I could start with 'p'
and plug him in my function 'g',
and then, I think you'll grant me,
I've got log base 10 of 'p'.
Now, I can plug him into my function 'f' 
and we know that we're gonna to get 'p' back.
Again, what does it mean
to plug this into my function 'f'?
Well, everywhere where we see an 'x'
in our function 'f',
we're gonna plug in log base 10 of 'p',
so we have 10 to the log base 10 of 'p',
and, again, that's equal to 'p', 'kay?
These are two of the properties that we discussed
in the previous section of 5.0,
and that's where they come from,
they come precisely
because these are inverse functions, 'kay?
And again, we can do a similar thing
for 'e' to the 'x' and the natural log of 'x'.
Again, we know that they're inverse functions,
so we know that they undo each other.
So, again, I'll start with the point 'p',
just so that we don't confuse all these 'x's,
and we could plug him in 'f' first ,
and I think you'll grant me
that 'e' to the 'p' comes out.
Now, if we take this and plug him in our function 'g'
we know we're gonna get 'p' back,
we know that it's going to be equal to 'p'.
Well, what does it mean to plug 'e' to the 'p' in 'g'?
Well, it means we're gonna take the natural log
of 'e' to the 'p',
and notice, again,
we have this similar property, 'kay?
One more time, let's go the other way.
We start with 'p', we could plug him in 'g' first,
and so we've got the natural log of 'p'.
Then we could plug him in 'f',
well, that means we're gonna
look at 'e' to the natural log of 'p',
and because these are inverses
we're gonna get back to where we started, 'kay?
So, in general,
let me say this,
and this is actually-this works for any base,
in general though, 'e' to the natural log of 'p' is 'p',
the natural log of 'e' to the 'p' is equal to 'p',
and similarly 10 raised to the log of the 'p' is 'p'
and, in a similar manner,
log of 10 to the 'p' is 'p', 'kay?
These are really, really useful things.
Alrighty, so,
now that we've establish the equivalence
and the fact that these guys are inverses
and, therefore, undo each other,
what I'm gonna do
is play around more with these properties
that we played with in the last section, 'kay?
So, let's take some examples.
Okay, so for my first group,
what were going to do is evaluate
without a calculator...'kay?
Again, what we're doing here
is going to be looking very similar
to, um, what we did in the previous section.
So, first let's start with this,
let's do log of the square root of 1000, 'kay?
Now, the way I like to approach these
is I see that this is a log base 10,
so what I'd really like to do
is get what I'm taking the log of
as 10 raised to some power,
and I claim we can do that.
So we've got log, 'kay?
Now, 1000 is 10 cubed,
and when you take the square root
that's the one half power.
Again, when you raise something to a power
raised to another power
you multiply the exponents,
and now we can use the fact that their inverses
and we could just say that this is three halves,
'kay?
Alright, let's take another example.
What if I have,
how about the log of the square root of 10?
Well, now I nearly have this written
as 10 to some power,
I'll just write that as 10 to the 1/2 power,
and so we get 1/2.
Okay, what if I have this example?
10...raised to the log 1, 'kay?
Now, using this property
we know this is equal to one.
Another thing you could do is you could say,
"Wait a minute, the log of one is zero,"
okay, I'm just showing you
that there's another way,
and it brings you to the same-same place.
We know that the longer one is zero,
and we also know that 10 to the 0 power is 1.
Alrighty, what if I want to look at, say,
the natural log of 'e' to the fifth.
Again, using the fact that these are inverses,
remember, you're taking five,
plugging him in your exponential,
taking that and plugging it into your log,
they undo each other, you simply get five.
Little anticlimactic perhaps.
What about 'e' to the natural log of two?
Again, these guys are inverses,
you simply get two.
Alright, what about this one?
The natural log of one over the square root of 'e'.
Now, we can actually do this
a couple of different ways.
Obviously, on any test
you're only gonna have to solve it once,
but I'm gonna show you both ways of doing this,
and again, show you that you get
to the exact same place, 'kay?
Using what we've been playing with
in this particular section,
what I'm gonna do is write this
as the natural log of the 'e'
to the negative one half power.
Now, how did I get 'e'
to the negative one half power?
First off,
note that this is one over 'e' to the one half,
so all I did was bring him up to my numerator,
and that got rid-an-and that, excuse me,
and that made my exponent negative, 'kay?
Now, I can use that property over there
that these guys are inverses,
and we're just going to get negative one, 'kay?
Another way that you could look at this,
using what we did in the last section,
is you could say you've got the log of a quotient.
So can't we write that as the natural log of one
minus the natural log of 'e' to the one half power?
The natural log of one is zero.
Boy, using those properties
we could even bring that one half down in front,
or we could just keep consistent
and say these guys cancel each other out.
So we have zero minus one half,
which gives us that negative one half.
So do you see how, when we use these properties,
you can do it one way, I can use it another way,
but we get to the exact same place.
It's a beautiful thing about math, one of the many.
Okay, now what I would like to do
is...rewrite these without using logs.
So, express
in terms
of 'x'
without our natural logs...'kay?
Again, we're just playing
with simplifying these things.
Natural log of 'e' to the two 'x' power,
these guys are inverses,
they cancel each other out, they undo each other,
and therefore you get two 'x'.
How about 'e' to the natural log
of three 'x' plus two?
Again, our input was three 'x' plus two,
we plugged it in our function,
we then took the output, plugged it in our inverse,
and we get back to where we started.
What about the natural log of one
over 'e' to the five 'x' power?
Well, first thing I would do
is I would write that as the natural log of the 'e'
to the negative five 'x' power.
Again, what I did
is I brought my exponential up to my numerator
because my exponent was positive down here,
yes, that did give me a negative exponent,
these guys are inverse functions,
they undo each other,
so we get back to where we started.
And last, but not least, for this example,
would find the natural log
of the square root of 'e' to the 'x'?
Well, first thing I would do, as I've done prior,
is I'd rewrite my square root
as a one half power, 'kay?
Again when you raise something to a power
raised to another power
you multiply your exponents,
and so we're left with one half 'x'.
Alright,
now, what I'm going to do for the next few examples
is as follows,
this kind of shows you
where we're heading with these things, 'kay?
For example,
suppose I have 1.45 to the power of 'x',
equals, say, 25, 'kay?
This was sort of our motivating factor,
and don't worry,
we're going to get a lot more in depth
with solving these sorts of equations,
but I'd like to get us going
with a few right now, 'kay?
We want to solve for 'x'.
Alright, well, the problem is
is that 'x' is up in our exponent
and we have no real way of getting him down.
This is where the beauties
of-beauty of logs come in, 'kay?
Now, you can either use the l-o-g log,
in other words, log base 10,
or you can use the l-n log, the natural log.
You're gonna wanna use one or the other, because that's what your calculator can compute.
Honestly, I'm a natural log type of person,
so that's what I usually go to
unless I have a base of 10,
if I have a base of 10
I'm totally gonna use log, okay?
So, in this case, notice I don't have a base of 10,
so what I'm going to do is use either one,
I'm just going to take the natural log of both sides.
Now, remember, equations are like kids,
whatever you did one you got into the other,
so, in other words,
if I'm gonna take the log of one side,
in order to keep quality,
I need to take the log of the other side
Now, you might ask, "Why do you do that? it looks heckuva lot more complicated
it looks heckuva lot more complicated."
Well, now we're gonna go back
to one of those properties of logarithms
that I had mentioned to you
was one of the most often used, 'kay?
Remember, when you have the log
of something raised to a power,
you can bring your exponent down in front.
So all I did from here to here
is use my property of logarithms
that say that I can bring my exponent down in front.
Now, and natural log of 1.45 is just some number,
the natural log of 25 is just another number, 'kay?
We want to know what 'x' is,
we want to get him alone.
Well, how do you undo multiplication?
You divide both sides.
So what I'm gonna do is divide both sides
by the natural log of 1.45.
On my left side they're gonna cancel,
that was the whole point,
and on my right side, I get the natural log of 25
divided by the natural log of 1.45,
and if you're at all curious
you can solve this on your calculator.
And, going to two decimal places,
we're gonna get about 8.66, 'kay?
And, you know,
you can at least check your answer somewhat,
I just plugged in 1.45 to the 8. 66 power,
and when I did I got 24.97, okay?
So, within roundoff error,
we did a pretty good job here.
Alright, now, let's take this example:
'e' to the 0.12x equals 100.
In this case,
I wouldn't even think to use my l-o-g log, 'kay?
Why, notice the base here,
you certainly can, mind you,
but notice the base here is 'e', 'kay?
So, whenever you have a base of 'e'
you wanna use your natural log.
Just like whenever you have a base of 10
you want to use your l-o-g log, 'kay?
Let me show you why.
If I take the natural log of both sides,
again, we want to get this value of 'x'
and we really don't have a way
to get to our exponent.
We have the natural log
of 'e' to the 0-to the .012x,
excuse me, I put my decimal point
in the wrong place there,
equals the natural log of 100,
again, whatever you do to one side of the equation
you do to the other side.
Now, because these are inverse functions
they undo each other.
Notice, my left hand side just becomes .12x,
and that's equal to the natural log of 100.
We can certainly get 'x' alone
by dividing both sides by .12,
so I claim that 'x' is equal
to the natural log of 100
divided by .12.
Now, I'm gonna compute that,
and to 2 decimal places we get 38.38, 'kay?
I want to show you something,
this is not something that I would do in practice,
but I want to show you
that you would get to the same place
if you chose to use log base 10.
If i took the l-o-g log
of both sides,
well this is kinda cute to do anyway, right?
'Cause check this out,
what would we do here?
We would bring our exponent down in front,
we would have .12x times the log of 'e',
We could leave this as log of 100,
but you guys, I can't resist,
isn't 100 10 squared?
These guys are inverse functions
so we just get two.
Alright, we want to get 'x' alone.
What's the problem with this left side?
Well, not only are we multiplying by .12,
but we're also multiplying by the log of 'e'.
So, in order to get 'x' by his lonesome
I need to divide both sides by that mess.
So, I claim 'x' is 2
divided by 1.2 log 'e'.
And if you plug that into your calculator
you will see
that you're going to get
the exact same thing.
Isn't that so cute?
I just love that.
Alrighty, so,
let us do one more example,
for now.
Let's suppose we have 48
is equal to 17 times 2.3 to the 'w' power.
Alrighty, well, what do we want to do here?
Again, we want to find the value of 'w', 'kay?
And yes, we do want to take the log of both sides,
we could do the l-o-g log or the natural log,
one is not easier than the other in this situation,
but the problem
is we don't have a exponential isolated, 'kay?
So, before you take the log of both sides,
what you want to do
is isolate your exponential, 'kay?
We're going to be solving, um,
certain equations like this
where we actually have an exponential
on both sides,
but one of the things you're gonna see
is we're gonna wanna put our exponentials
on one side and our numbers on the other, 'kay?
So, the first thing that I'm going to do
is divide both sides by 17.
So I have 48 over 17 is 2.3 to the power of 'w'.
Now, we could leave it is 48 divided by 17,
or you can get a decimal approximation,
either way.
What I'm gonna do now
is take the log of both sides,
again, l-o-g log or natural log,
you will get the same thing.
So, I have the natural log of 48 over 17,
is the natural log of 2.3
to the power of 'w'.
My next step is to bring my exponent down in front,
so on my left I have the natural log of 48 over 17
and on my right I have 'w'
times the natural log of 2.3.
in order to get W alone
I'm gonna divide both sides
by the natural log of 2.3.
So, I have the natural log of 48 over 17
divided by the natural log of 2.3
is equal to 'w'.
It doesn't look pretty,
but our calculators can certainly compute this
without a problem.
And what we're going to get is as follows:
approximately 1.25.
Don't worry,
we will be playing more with these very soon,
but I think that that's enough for this section,
thank you.
