This is the first of several videos
on logarithms and their algebraic properties
First, what is a logarithm?
What do we mean by log?
So... say that the logarithm of x equals some number, "a"
if 10 raised to the "a" power equals x.
That's the definition of a log.
It's a slightly indirect definition,
but that's the definition that I always go back to when thinking about
what logs are.
There's another way to say this.
You can basically write this out in a sentence.
Let me do that.
Another way of saying this as a sentence is
that log x is that number which,
if you put in an exponent above 10,
gives x.
In other words, 10 to the log x, is x.
This tells us that logarithming, taking a log
and then exponentiating, they undo each other
They are actually inverses of each other.
One does something the other undoes something.
Logs undo exponentiation.
If you log something and then exponentiate it
you're right back to where you started.
This is the basic equation.
Really these two say the same thing.
That we'll use when we start to think about logs
Let's do a few examples using this definition.
Let's see... log of 100.
What would that be equal to?
I claim that it's equal to 2.
Why?
Because 10 to the 2 equals 100.
2 is that number which, if I put it up in an exponent for 10
gives me 100. So, log of 100 is 2.
We could do another example.
Log of 100,000... what does that equal?
I claim that equals 5.
Why?
Because 10 to the 5 is 100,000.
OK. Let's do one more example.
Maybe I'll just write this again.
That's the thing we remember for logs.
So what about
log of 500?
I'm looking for some number which, if I
put it in the exponent, gives me 500.
Here, without a calculator, I can't do that exactly.
But I can figure out some stuff... so
10 to the 2 is 100.
10 to the 3 is 1000.
so 500, because it's between these two numbers...
the log of 500 must be between 2 and 3.
Let me write that: log of 500 is between 2 and 3.
If you needed an exact value,
you could look it up on a calculator.
So, I can do that. 500... log...
About 2.699
It turns out that log of 500 is about 2.699
Why?
Because 10 to the 2.699 is going to be about 500.
I can test that out on a calculator, too.
10 raised to the 2.699... sure enough, is about 500... 500.03
Alright. This down here, you would need a calculator to do.
To figure out an exact value,
or a closer to exact value.
But without a calculator, you should be
able to know that log 500 is between
2 and 3. Because 10 to the 2 is 100
and 10 to the 3 is 1000.
In the next couple of quizzes,
you'll practice thinking about logs
using this basic relationship
and along the way, you'll discover
values for a few special numbers..
logarithms for a few special numbers.
So, give the quizzes a try, and if any of them are confusing
be sure to check out the solutions.