Welcome ladies and gentlemen.
And what I'd like to do is show you
how to evaluate logarithms without using a calculator.
Now, obviously for a majority of these--
or actually for all of these, you
could type them into your calculator using
the change of base formula unless your calculator can
do different bases.
But none of these I chose base 10 or I didn't
do any natural logarithms.
But for all logarithms you can always just
type in your calculator.
But I guarantee I could do all of these
as fast as it would take you to pick up your calculator
and plug them in.
And it's just because using the understanding of logarithms
has helped me been able to do all these problems,
as well as negative exponents.
So that's one thing I want to write into here is-- well,
there's a couple things.
First of all, remember we have a logarithms base
b of x equals y.
We can always rewrite that in logarithmic form as b to the y
equals x.
The other thing I want to remind you
is if I have an exponent in the denominator,
I can always rewrite that as an exponent in the numerator,
but just using it with a negative power.
You can put that over 1 if you wanted
to but we don't really need to.
So basically, by knowing these two understandings,
I can do all these problems without a calculator very,
very quickly.
The other thing, actually, we also
want to know is using the one to one property
is sometimes helpful even though we're not solving,
we're just evaluating.
A lot of these, it is very important to understand this
because I think I'm going to do some difficult problems.
So if we have an exponent equal to another exponent
where the bases are exactly same,
then their powers are same.
So we can set their powers equal to each other.
OK, now, you can see that these are just expressions
that we want to evaluate.
But our basic mode of thinking is, again,
to find what their value is.
Now, remember when we're looking at a logarithmic equation--
for instance, log base 3 of 9-- we
know that what that answer is going to be is 2.
Because what a logarithm says and means is 2 raised to-- oh,
I'm sorry, 3 raised to what power is 9?
2, and if you were to rewrite that in exponential form,
it would look like this.
3 squared equals 9.
So again, that's writing it into my exponential form.
3, the base, raised to what power gives you 9?
That's what our logarithm is basically asking us.
3 raised to what power is 9?
2, and you can always think about rewriting it
in exponential form to kind of check your answer.
So, in my first example I kind of tried
to think of one that's going to be fairly simple.
This is basically stating us 5 raised to what power is 25?
Well, I know that 25 is a square number.
And I know that 5 squared is 25.
So therefore the answer here in this case,
it's just going to be 2, OK.
Now, the next one is 3 raised to what power is 81?
Now again, without a calculator, you
need to kind of start knowing the powers of 3, 4, 5, 6, 2.
So what I would start doing, is if you're not
really familiar what this would be, start raising 3 to powers.
3 to the first power is 3.
3 squared is 9.
3 cubed you might have to type in your calculator,
but eventually you'll understand it's 27.
3 to the fourth power you might have to type in your calculator
originally, but you'll see that it's 81.
So that means 3 raised to what power is 81?
Well, the answer is equal to 4, OK.
Now, in the next example what I'm going to do
is I'm just going to kind of think
about it in another format.
Now, you could do the exact same process I did over there,
and just start listing 2 to what power, you know, equals that.
But I also want you to think about this as maybe
possibly an equation.
Here's an expression.
Now, again these are expressions.
We're just evaluating.
We're not equations.
But really what an expression is is
you're saying it's equal to what value?
So I'm actually going to replace that as an x.
And then what I could do is using,
by rewriting this as exponential form,
I can rewrite this as 2 to the x equals 64.
So therefore again, if you're having trouble understanding
what the logarithm is, set it equal to x and then
rewrite an exponential form.
2 raised to what power is 64?
And we'll get to the one to one property here in a second.
But what we could do, again, is just think about this.
Well, if I know that 2 to the first, 2 the second,
2 to the third, 2 to the fourth, 2 to the fifth, 2 to the sixth.
2, 4, 8, 16, 32, 64.
So 2 to the 6th power.
So therefore, I know that x, or that's equal to, 6, OK.
Now, where that variable kind of comes in handy
though is in a problem like this,
8 raised to what power is 4?
Oh, I don't know 8 raised to 8.
8 raised to the first power is 8,
so that's already larger than 4.
So how would I do this?
Again, if you use the method that I did in the last one,
you set it equal to x.
Now, rewriting it in exponential form,
gives me 8 to the x equals 4.
Well, now I can use this one to one property that I stated.
And by using the one to one property, what
I notice is 8 and 4, they have to have the same base, right.
Well, 8 and 4 do not have the same base.
But you can use the same base of 2.
Because 2 squared is 4 and 2 cubed is 8.
So then what I do is I rewrite this as 2 cubed raised to the x
equals 2 squared.
Now, that they have the same base,
I can just set the powers as equal to each other.
So I have 3x is equal to 2 so x equals 2/3, OK.
All right, so now we're going to get into some fractions.
Again, into the same respect, main important thing
that I can do is I can rewrite all fractions-- again,
you could think of this as 4 to the positive first power--
you can think of all fractions as rewriting them
with negative powers.
So now I can rewrite this as log base 4 of 4
to the negative first power.
Instead of it being to the first power,
you can write it to the negative first power.
Now, there's another rule of logarithms
that you should remember.
Log base b of b equals 1.
And log base b of b to the x equals x.
Well, if I'm taking the log of the same base of 4, raised
to the negative 1 using this rule of logarithm,
my solution is just going to be a negative 1, OK.
This example might be a little bit confusing.
I have fraction and fraction.
I could rewrite them as negative powers.
But again, what I'm seeing here is probably
rewriting them in exponential form would probably
make more sense for me.
So I'm going to do 1 over 6 raised to the x equals 1/36,
OK.
Now, I use my negative powers.
So I'll do 6 to the negative first, raised to the x,
equals 36 raised to the x.
OK, now I want to see-- oops, that's negative first power,
isn't it?
OK, now I want to see, can I rewrite 6 and 36
to the same base?
Well, yeah, 6, 36 squared is-- 6 squared is 36.
So I get 6 to the negative x when I apply the power rule.
Remember, the power rule states if you
have an exponent raised to another exponent,
you multiply them.
36, I can rewrite as 6 squared times negative 1,
which is raised to the x.
Again, you're just going to multiply these.
The 6 is divided out using that one to one property.
So I have negative x equals negative 2.
What happened here?
Why do I have an x up?
I don't have an x over here?
Why is there an x over here?
I think I was going crazy.
So that equals negative 2 divide by negative 1,
divide by negative 1, x equals 2.
OK, in this case, basically again what
I'd like to do-- OK, basically in this case is,
again, I could use my negative powers like I did over here.
So I can rewrite this as log base 2 of 8
to the negative first power.
Now, again that's still not really,
you know, helping me out, right?
But again, think about this as using this rule.
Can I rewrite 8 as a base 2?
Because if I can rewrite it as a base 2.
It's the base raised up of the same base.
And yes, I can.
Log base 2 of 8 can be written as 2 cubed raised
to the negative 1, which is log base 2 of 2 to the negative 3--
because you multiply those powers.
So therefore, my answer is just equal to negative 3.
OK, and last but not least, I have
log of 1/3 raised to the 27th.
Again, what I would do in this case
is I would set this equal to x and I'd rewrite it
as an exponential equation.
Then I'd use my negative powers, 3 to the negative 1,
x equals 27.
Then what I want to do is I say, can I
rewrite 3 raised to-- 3 as-- can I have
3 and 27 with the same base?
And yes, 3 to the first power is 3.
3 to the third power is 27.
So therefore, I just need to rewrite 27 as 3 cubed.
Therefore, negative x equals 3, x equals 3.
So there you go, ladies and gentlemen.
That is how you evaluate logarithm expressions
without using a calculator.
Thanks.