All right in this video. We're going to talk about logarithmic functions
So everything you ever wanted to know about logarithms
but we're afraid to ask we're going to try to cover in this video, and there's are quite a few properties, so
You know it may take a few minutes here to get through it
if you're looking for
Information about derivatives of logarithms that's on a separate video
so is
logarithmic differentiation
So if you're looking for those two things you're looking in the wrong place right now these are just going to cover the basic
basic Properties of logarithms
[so] [one] thing the graph of a logarithm
so a logarithm looks like y equals [log], and then we call it a base a
of our variable x and
This base a can be any number greater than zero we don't let it equal one
But the most important case is going to be when this base is greater than one and if this base is greater than one
This will be a graph of a basic loGarithm
so this is the point 1 0 so all logarithms will go through 1 0 it gets close to the
[y-Axis], but it never actually touches it and then it keeps growing
But it definitely kind of tapers out it stops growing very quickly
So that's the basic graph of a logarithm
Ok so what the heck is a logarithm. Well. It's kind of a difficult question to answer I suppose
Computationally the way that you'll want to deal with logarithms
Suppose I have log
base a of x
Equals y, I?
Can turn this into an equivalent exponential and you write that as a to the y?
equals x
So this is definitely one of the first little rules that you'll want to commit to memory for a logarithmic function
So let's do a couple examples of these here real quick
suppose somebody said find
log base 10 of 1000
All right, well so log base 10 of 1000. This is just some [number]. I'll call
[it] x I don't know what it is
but notice this pattern on
Going from a logarithm. It says if you take this base a and raise it to that power y
you'll set it equal to x
So equivalent equivalently. It says I can rewrite this as
10
raised to the power of x equals
[1000]
Well don't say it's 10 times 100 because we're not multiplying you have to think about exponents
So tend to what power is a thousand well 10 squared is a hundred
10 cubed is a thousand so it turns out that x has the value of 3 or
We could say likewise that log base 10 of 1000 is
a Fancy way for writing the number 3 and
Logarithms to me. You know almost anytime I see them. I'm definitely thinking about the equivalent exponential form in my head
Let's do one more here
Suppose I have log base
5 of
1 over 25
so same way this equals some number x
Well, I can rewrite it as 5 to the x equals 1 over 25
But notice I can write 1 over 25 as 1 over 5 squared
[right] 5 squared is 25
and
Remember through properties of exponents I can bring this five to the numerator
by making the exponent negative
So if you look basically at the left side, it says five raised to some power
Has to equal five raised to the negative two
Well in that [case]. We must it must be the case that the powers are equal. So we must have that x equals negative two
Okay, so this power up here has to equal this number
so x equals negative two and
We're finished so again log base 5 of 1 over 25 is a fancy way [for] writing the number Negative 2
All right, so kind of the first basic properties you'll want to know with logarithms
The next three important properties that are worth remembering
Are going to be kind of I don't know always considered consider them kind of the algebraic rules of logarithms?
[so] more rules
It says if you have a logarithm
You know regardless of the base it?
Says if we have two things being multiplied you can break up the logarithm
But it doesn't become the logarithm of one times the logarithm of the other
it becomes the logarithm of
One thing plus the logarithm of the other thing
likewise if you have division
Well multiplication goes to addition. You may be able to guess that division becomes
subtraction
so you'll get log base a of x
minus log base a of y and
the other property says
If we have something raised to a power in this case x to the N
it's a there's a rule that says this power can come out front as a
Basically is a factor. You're going to now be multiplying by that power and notice. It's now gone on the right side
so a few more rules that you're going to want to remember with logarithms [if]
There's a plus or a minus inside the parentheses you can't do anything with it. You just have to leave it
[just] like it is so the properties only apply when you have multiplication division or one of these powers
Obviously, you know since there's an equal sign
you can go from the right to [the] left as well if you have a logarithm of one thing and
A logarithm of another thing if they're the [same] base in this case a well
You can rewrite the addition as a single loGarithm as multiplication
So let's do one of these
Suppose we want to expand
let's do Log Base 2 of
x cubed
times y over
z squared
Well, you can do this in one fell swoop. I'm going to do it in a couple steps
Again, it says if you have whatever's on top you make it log base 2 of that [-] log base 2 [of] the bottom
so I'm going to get log base 2 of the top x
cubed times y minus
log Base 2 of Z squared
And I have to be careful here. I can't pull the [3] out because everything's not being raised to the 3rd. Just the x
but now I do notice that I have multiplication so I can break up this first one as
addition
and
I am going to use this property on [my] second one over here
It says if you have something raised to a power you [can] pull that power out front, so I'll get minus 2
log Base 2 of Z and
The last thing I'm going to do is pull out this 3
So I'll have 3 times log [base] 2 of x plus
Log Base 2 of y
minus 2 times log [Base] 2 of Z and
That's now nice and expanded
The instructions could have started you with the thing at the bottom and they could have said well rewrite this as a single quantity
Basically you would just reverse the steps you route you would take any
Coefficients and put them up as powers so all the [coefficients] would go up as powers
And then if things are separated by a plus again they have to be the same base
and they all in this case are if things are separated by a plus they would become multiplication and
Then if things are separated by a minus you just end up dividing
So it would be pretty simple. [I] hope to go back the other direction as well
if this is the first time you've seen logarithms, or
It's been a while. It may [not] be quite so simple
it can definitely be [a] little confusing so
But it's one of these things once I think you do a few of them and get a handle on them
You'll be you'll be okay
Just something notationally. That's very important
And I'm not going to do any examples of these just recall so this is just notation
when somebody writes
Log, base E and your call a is just the number two point seven one and there's more
to it
It turns out that e is kind of a very special base for one reason or another
so what they do is will rewrite log, base E as
Ln of X
or the natural logarithm sometimes people pronounce this Lin x
Natural logarithm of x Ln of x so all of those mean the same thing
likewise if somebody rate writes just log of x it's
understood that that means log base 10 of x
Okay, so just some notations. [I]
Guess I should put my log base 10 here
But whatever so these are the ideas so log of x is log [base] 10
Ln of x means log base e of x so just something to keep in the back of your mind
when you're working with these things
Probably the last important property to really talk about with logarithms are
What are known as cancellation laws?
Okay, so the cancellation laws
and
What this says is these again tie into?
exponential functions it
says if you have an exponential a to the x and
You put that inside of a logarithm
So log base a of a to the x and notice I've got the same base that again is crucial
this is going to be equal to x
likewise if I have a logarithm
log base a of x and
Then I sort of put that inside an exponential so again log base a of x is think about that as being a number
and I take a and raise it to that number that's going to also equal x and
These rules are valid this first [ones] valid for
all
numbers that you can think of and
the second case this is only going to be valid for x greater than zero and
Again, there's some technical reasons for this
Mainly the idea is if you [think] about log base a of x if you go back to the domain
If you look at the graph notice the domain [of] a logarithm is all x is greater than zero
So we have to start off with those numbers. We're kind of in trouble
if you start off with an
exponential function a to the x well
exPonential's have domains all numbers notice you get positive numbers out and
Again those are numbers that you can put inside of a logarithm
The main place you're going to use these cancellation laws are when you have to solve [equations]
That involve either exponential functions or logarithmic functions
So for example suppose somebody said solve this one. I've got [two]
Ln of X
equal to one
Well typical thing I'll do in these problems
You know if it was just two times x I would divide both sides by two and I would be done
I'm going to do that
Same thing here as well. I'm going to take Ln of x and I'm going to get it by itself by dividing by two and
Again, here's where I'm going to use this property
Find a piece of paper here real quick
This is where we're going to use this property
[well] first off. I'm [just] going to rewrite the natural logarithm as log base e of x
So again log base e of [x] this is equivalent to one half so now I'm going to use this property
It says whatever the base is you raise both [sides] to [that] to that power or excuse [me] to that base?
So I've got log base e. I'm going to take E and raise it to the left side
Well if I take E and raise it to the left side
I have to take e and raise it to the right side and by this cancellation rule
It says all you're left with is the stuff next to the logarithm
So now it says I'm left with just x on the left side and that equals e to the one-half
which I could also write [as]
Square Root of E
so square root of e is going to be the solution to this original equation that I started off with
let's do maybe a couple more of these here and
Then we'll be pretty much done with logarithms
Suppose I have e to the negative x equals
five
Well before [I] wanted to get rid of a logarithm notice
I wanted to get rid [of] a logarithm, and I did e to both sides
Now I want to get rid of an exponential
I'll take the logarithm
Using the same base
of both sides
and
Again if I refer back to my cancellation laws
It says if these bases match up they cancel out, so I'm just going to be left with negative x on the left side
Well, log base e of five that's some number and without a calculator. I just don't know it
I'm going to rewrite log base e as natural logarithm
Remember these are equivalent notations, and if I multiply [both] sides by a negative
I'll get simply that x equals negative Ln of five and
That'll be my answer
Looking for one more good example here
Maybe just one more very basic example
Suppose I have 5 raised to the x [minus]
3 equal to 10
Well again the bases have to match up, so I'm going to use log base 5 on the left side
I'll use log base 5 of the right side
Again these two symbols match up, whatever's upstairs. That's what you're left with
Log base 5 of 10 I don't know that some number
I'm just going to leave it like [it] is because I don't know what it is
now to solve for x I'll just add 3 to both sides and
Again, I'm finished. I've got my answer
You could use these cancellation laws a couple times, so this will be our grand finale here
suppose I have Ln of Ln of x equals 1
Well again, this is log, base. [E], so if I raise e to the first side to the left side
I'll raise e to the right side
Again, it says if you have e and log base e whatever's next to it
That's what you're left with so I'll get Ln of x equals e to the first
But I can actually do this again. I can take e and raise it to both sides
and
Again now on the left side. I'm just going to be left with plain, Ol x
Equals E raised to the power of E, and that'll be my solution
So here are some basic [examples] and some basic properties [of] logarithms again
[you] know they're definitely harder examples with logarithms
Trying to give you a flavor of just some of the very beginning ones some of the more straightforward ones
Again the things to remember make sure you know the graph of a natural logarithm
Make sure you can convert from a natural logarithm to exponential notation
Make sure you know those properties involving multiplication division and powers
inside of a logarithm and
make sure that you know the cancellation properties of logarithms, so
again quite a few things to [remember], but if you do know those things
I've managed to get through a whole career [in] math just knowing those things
So I think if you know that stuff [you'll] should be in pretty good shape
[and] I think again, you know logarithms are something that you don't encounter that often so they can be a little intimidating
But I think once you do a few examples and get a hang on get a handle on them
They won't be that bad so again if you need some more help feel free to visit my website just math tutoring [komm]
Lots of videos [on] there again if you need to see derivatives of logarithms
Or logarithmic differentiation again. Those are kind of two
[different] things
Definitely dig around there. I've got some video showing quite a few examples of that stuff. So good luck. I hope this helps
