Welcome, ladies and gentlemen.
So what I'd like to do is show you
how to evaluate a logarithm without a calculator.
Now, what I'm going to do for this video
is do some logarithms that are pretty simple.
And I think as long as you understand
what a logarithm represents and what it means
and what it says when we have a problem,
then I think you'll be OK for this video.
So basically, when we have a logarithm-- for instance,
let's just use log base b of a-- what exactly this is saying
and what exactly we're expecting to evaluate
this-- is what we're asking is b raised to what number is a?
So a lot of times, we write that in our exponential form, which
would say, b raised to what number
is-- which would be equals-- a?
Now a lot of times, b raised to what
number is a-- that's how we can write it as an equation.
So let's just look at something that's very simple.
I don't think I did this.
Yeah.
OK.
So we have log base 2 of 4.
So what this is saying is 2 raised
to what number gives us 4?
So 2 raised to what number is 4?
Well, hopefully, you guys know that 2
multiplied by it-- raised to the second power-- is 2.
So therefore, the answer for that would be 2.
Now, we're not going to go through all those details,
but a lot of times, that's what we're basically saying.
And for all these problems, that's
exactly what I'm going to be thinking about,
is what I'm going to raise.
So in this example, this is saying 3 raised
to what power gives us 9?
Well, we know that we can rewrite 3
to the second power equals 9.
So 3 raised to what power gives us 9?
So our answer we evaluate is just going to be 2.
In this example, 5 raised to what power gives us 25?
Hopefully, you know that 5 squared equals 25.
So therefore, we say 2 is our answer again,
just like the last one.
So I tried to do some kind of simple ones here.
And then we get into 4 into 64.
And some of you might not remember 4 raised
to what number gives you 64.
So if you get stuck with a lot of these, what I would do
is just start listing out 4 raised
to all these different powers.
So 4 to the first power is 4.
4 to the second power is 16.
4 to the third power-- now you might have to use a calculator,
but 4 times 4 is 16 times 4 again-- guess what-- is 64.
So therefore, 4 raised to what power gives you 64?
The answers is 3.
Over here, again, 2-- you might know that 2 squared
is equal to 4.
You might remember 2 cubed is equal to 8.
But you might forget, you might not know if it gets up to 32.
So let's, again, just write them out.
So again, it just takes a little bit of time.
But once you get used to working with these numbers everything
will start coming together, and you'll
start remembering a lot of these.
And you won't have to write them out every single time.
So remember 2 to the fifth is 2 times 2 times 2 times 2 times
2.
You could also type it in your calculator.
But you see 2 raised to what number gives you 32?
Without a calculator, I can determine that's going to be 5.
Over here I have log base 3 of 81.
I went to 3, so I didn't continue that.
So let's do 3 to the first power is 3.
3 squared is 9.
3 cubed is going to be 27.
And 3 to the fourth power is 81.
And again, you might just have to type in your calculator.
Type in your calculator 3 cubed, 3 to the fourth,
until you really get really good at these.
10-- now this is always a great one.
Oops, I don't think I was supposed have a 10 have there.
I was supposed to tell you, remind you,
that when you have log of, looks like, 10,000 here,
remember that has a base 10.
So again, 10 to the first power is 10.
10 squared is going to add an extra 0, so that would be 100.
So 10 cubed is going to be 1,000.
Well, if I have to do one more-- therefore,
that's going to be 10.
That's going to be to the fourth power.
I'll do this one first, then I'll get to these last two.
Here I have log base 6 to the 216th.
Again, I don't remember 6's.
I tried to stick with the 2's, 3's, and 4's.
Those are the most common.
And 5-- but I didn't do a 5.
I did one 5.
I probably could have done a harder one.
But anyway, 6's.
6 to the first power is 6.
6 squared is 36.
6 cubed might have got by you-- I don't know what's 6 cubed is?
Well, guess what?
It's 216.
If you type in your calculator, you'll be able to see.
So again, what the logarithm is saying
is 6 raised to what power gives you 216?
Well, that is 3.
Now, the next two are ones that, again,
as long as you say them out loud, as far as
what the logarithm means, I think they'll make more sense.
7 raised to what number equals 1?
Well, it kind of gets tough here because if you did 7
to the first power, that equals 7.
If you did 7 to the negative first power, that's 1 over 7.
So what number, when we raise it to a power is going to equal 1?
Well, I just kind of misspoke there-- not really misspoke.
But the answer, again, is going to be 0
because any number-- any number-- raised to the 0 power
is going to equal 1.
Always.
And this is the last one.
I have log base 11 of 11.
So basically, I'm saying 11 raised
to what number gives you 11?
Well, we obviously show from this first one
any number raised to the first power
is just going to be itself.
So therefore, the answer here is 1.
I guess I could've done one of those.
No, I didn't do that.
Oh well.
So there you go, ladies and gentlemen,
that is how you evaluate logarithms,
basic simple logarithms, without a calculator.
Thanks.