HELLO, Mr. Tarrou:D
We just finished talking about exponential functions and graphing them.
That leads us naturally into logarithms.
I would like to, if you have a calculator handy, what is the log of 100.
Can you find that button on your calculator?
It is equal to two.
What is a logarithm? The book says that the form of a logarithm is y equals base b of x
where x is greater than zero.
And b has to be greater than zero. Both x and b must be positive.
Why does the log of 100 equal to 2?
Well, if you write log the standard base is ten.
So, log base 10 of 100 is equal to 2.
I always use this example when I start logarithms because it is one that we can do in our head.
You might recognize those numbers and go ok, 10, 2, 100...yeah 10 squared is equal to 100.
So what you can now see is that
I have an equivalent equations written in two forms.
I have one written in log form
and exponential form.
This very simple example that I use all the time in my classes,
to show my kids what is going on with logarithms.
It is nothing more than, when you take a logarithm
what you get out, look at this form that we are comfortable with,
we got out an exponent.
The base of the log ended up being the base of our exponent form.
The log base 10 of a 100 is 2 because 10 squared is equal to 100.
So, this is our base when we are looking at
equations that are in exponential form.
When we want to go from log form to exponential form the base of the log is the base of our exponent.
What is this 100 that we are logging? In exponential form it is the answer.
The log base 10 of 100 is 2, is equivalent to equation 10 squared is 100.
Log form...exponential form.
So keep this in mind. I am going to erase this and write how I like to write logarithms.
This will help you go back and forth between log and exponential form.
The log of a base...and then you have an answer
and what you get out of a logarithm is an exponent.
So lets use this pattern, and please write you base down here as a subscript,
don't write it up here in the main line because it is going to make it look like a lot of multiplication or
You may be writing log base 2 of 8 and it is going to look like log of 28. This will confuse your teacher and yourself.
Lets use this now to go back and forth between exponential form and log form and see
if we can evaluate some basic log expressions.
We have 4 equals the log base 2 of 16 which is
a true statement.
We are just getting into logarithms and we are going to convert it from log form which we may not be familiar with into exponential form.
Well this is log form and when we go into exponential form it is base, answer, and exponent.
So the base of the log is going to be the base of our exponent.
What do you get out of a logarithm? Just keep chanting that.
What do you get out of logarithms? Exponents.
What you are taking the log of in exponential form is actually the answer.
So, 4 equals log base 2 of 16 becomes 2 to the fourth equals 16.
Lets take a look at another example.
How about 3 to the negative 3rd equals 1 over 27.
Well is that actually true?. It is true.
These are all true statements.
This is now in exponential form.
I do not see a log function in here.
Remember when you have a base to a negative exponent, to get rid of that negative exponent you flip the base.
So this is one third to the third power.
This does equal 1 over 27.
Now to write that in log form,
Log...what is the base of the exponent?...
The base of your exponent is the base of your logarithm.
Then the answer. What is your answer in exponential form?
That is what you are going to be taking the log of.
So it is going to be the log base 3 of 1 over 27.
What do you get out of a logarithm? You get an exponent.
So negative 3.
You might be noticing...let me put this back up here again...
Log of a 100 equals 2.
and 10 squared equals 2.
You might be noticing that log function..you are undoing...
I have a hundred, I apply the common log, and I get 2.
Here I take 10 squared and I get back to 100.
I am doing something. I start with 100, I do something and get 2
I make that 2 an exponent of 10 and get back to 100.
Exponential form and logarithms, those are inverse math operations.
It is like addition and subtraction, multiplication and division.
If I do 10 times 2...I said addition and subtraction and I am doing multiplication...
10 time 2 equals 20, and 20 divided by 2 gets you back to 10.
Multiplication and division are inverse functions
5 + 2 is 7 and 7 - 2 gets you back to 5.
Addition and subtraction are inverse functions.
A couple of expressions going from log form to exponential form
and exponential form to log form.
Lets take a look at this expression.
Lets try to evaluate log base 4 of 64.
You might go, "well I tried that and checked the back of the book and it did not work" or "my calculator does not do log base 4"
Most calculators only do log base 10 (common log) or log base e called Natural Log. I have not talked about Natural logs yet.
What does this equal. I don't know and I like to put this extra part in that my textbook does not.
..when they give you these questions and ask you to evaluate them.
Write out that whole equation so you can see base, the answer, and what is going to be the exponent
when you write your log function in exponential form.
So this is going to be 4 to the x
equals 64.
And if you are in the first logarithmic section of your textbook
they are going to giving you problems you can do in your head, or without the aid of a calculator.
So four to the what power does equal 64?
I think four to the third equals 64.
4 and 4 is 16, and another 4 is...yes...it is 64
So, x does equal 3.
Notice that I wrote it so the bases were the same,
so the exponents had to be the same?
That is something you will be doing a lot when you solve equations that involve logarithms.
Or solving equations that don't need logarithms to solve.
GOOD...ALRIGHT!
Lets take a look at this question.
Log base 5 of 1/5.
What does that equal to?
I don't know.
But I know if the textbook gives me a log function I am probably not going to be able to do it
because again this has a base of 5 and most calculators won't be able to do, and INSPIRE will, but most calculators won't.
So lets write this log function in exponential form.
Base, answer, and what do you get out of a logarithm? An exponent.
This is 5 to the x power equals 1/5
Can I write this so that it has the same base as over here?
Can I move the 5 up to the top, in the numerator like it is in the left hand side?
I can if I allow it to have an negative exponent.
And so yes, log base 5 of 1/5 is equal to negative 1.
Or, what exponent will allow 5 to become 1/5?
Well that is negative one.
How about...
log base 7 of 7?
Now you your textbook might have said
when the base of a log equals the base of what you are logging that they are always going to cancel out.
..and that this would equal 1.
Let me explain why it does equal 1.
Lets say that I did not read the book and I do not know what log base 7 of 7 is.
Ok, so I am going to fill up the expression with base, answer, and exponent.
and convert my logarithm into exponential form.
What do you get out of a logarithm? an exponent.
So that is base, the exponent of that base
and then what is the answer.
Again these names are for when you are converting log form into exponential form.
So seven to the x is equal to seven.
Well, this had an exponent of 1
So seven to the x equals seven must mean that x is equal to 1.
And, if that happened to have been a 2
then it would be 7 to the x equals 7 to the second power
which would mean that x would equal 2.
What about this problem.
I have not written a Natural log up here yet, so lets see at least one of those.
The natural log of e to the fifth.
What does that equal to?
Well Mr. Tarrou, you haven't even talked about the idea of ln yet.
Lets explain that.
Natural log, ln, is log base e and I am going to take the natural log of e to the 5th.
Do you see these like bases starting to stack up? You might know what is going to happen now.
e is like pi. pi is approximately equal to 3.14........
e is approximately equal to 2.718.......that one runs on as well indefinitely.
This is a number that shows up in electrical engineering, and nature, compound growth, etc.
I don't want to derive where that comes from, but it is a standard numerical value.
So, Natural log is just log base e
and you can see we have these stacked bases and you might think they will cancel and you would be right.
I am going to write this log form into exponential form so we can see
the base, the exponent, and the answer.
So, base...exponent...and our answer.
Well look, I have a very simple equation. Both sides are equal, with the same base on both sides
so our exponents must be the same. Or, x equals 5.
You will see this enough times where you will say oh those bases match up lets drop it and say that x equals 5.
Alright!
Lets see if we can squeeze in another example before my 13 minute alarm goes off.
10 raised to the log base 10, not that I need to write for common logs, of 3
Your textbook may just say these cancel out...I don't need to write that 10 but I am going to leave it there.
..and they will say it equals 3 but lets figure out why.
Lets say I don't know what this expression equals.
Now this equation is in exponential form, not log form.
So, lets write it in logarithm form.
Log, what is the base of this exponent?, 10
What is the answer we are getting from this expression? The answer is x
What is the exponent?
Well this is all the exponent, log base 10 of 3.
Do you see how I am getting Exponent, and the Answer from the exponential form?
Now look, log base 10...same function and same base so x must equal 3
Or, as your textbook will show you
When you have a base and a log base stacked and touching, they will cancel if the bases are the same. And yes x does equal 3.
I am Mr. Tarrou. Thank you for watching, and go do your homework!