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Hello.
Welcome to the videos on logarithms.
What is a logarithm?
A logarithm is an exponent.
That might not make sense now, but after we have worked a few problems, I
hope you see this relationship.
We're used to this relationship.
If we see a to the b power is equal to c, well, if I rewrite that
relationship in logarithmic form, I will have log to some base, which in
this case is a.
Log base a of my value, which is c, is equal to b, which is my exponent.
So that's our relationship in log form.
And this is the relationship in exponential form.
What I often do in class is to write something like this--
base, value, equals exponent, because what do we see?
A base to some exponent is equal to a value.
So hopefully you can keep that relationship in mind.
As we work these, it's important to remember this strategy of prime
numbers with exponents.
Convert any large numbers to be a prime number with exponents
whenever you can.
And that will help you simplify your work.
I'm supposed to evaluate this logarithmic expression.
And let me tell you how I read this.
I read this log base 5 of 25--
log base five of 25.
So let's evaluate this.
It's in log form now, so let's rewrite this as log base 5 of 25 is equal to
some exponent.
I'm going to call that exponent x and rewrite this in exponential form.
We have 5 to our exponent is equal to my value, which is 25.
What's 25?
25 is equal to 5 squared.
So if 5 to the x is equal to 5 squared, what does x have to equal.
2.
So log base 5 of 25 is 2.
That one's pretty simple.
Let's evaluate another one.
What do I say here?
Log base 9 of 1/81.
And that's equal to x, some exponent.
So I want to take my 1/81 and rewrite that.
My base is 9.
I can rewrite 1/81 as a power of 9, can't I?
1/81 is 1 over 9 squared, or 9 to the what?
Negative 2 power.
So I can rewrite this as log base 9 of 9 to the negative 2 power is equal to
x, or rewrite it in exponential form, 9 to the x power is equal to--
what's the value?
9 to the negative 2 power.
And our bases are the same, so our exponents have to equal.
So in this case, x is equal to negative 2.
So our answer, when we evaluate, is negative 2.
Let's do some more.
Now, we're going to solve an equation that is in log form.
And in order to solve equations in log form, we are going to--
here's our strategy--
rewrite in exponential form.
So let's rewrite this in exponential form.
What's my base?
10.
What's my exponent?
Negative 2.
What's value?
x.
x is equal to 10 to the negative 2 power, or 1 over 10 squared, which is
equal to 1/100.
So x is equal to 1/100.
And here we have another one.
Once again, what's our strategy?
Rewrite our log in exponential form.
So what's my base?
x to the negative 4 power is equal to 1/81.
Now, I want to rewrite my 1/81.
Instead of as a base of 9, this time let's go all the way
down to prime numbers.
1/81--
that's 1 over 3 to the fourth power, or purely as a