Hi, let's take a look at properties of logarithms.
As I go through them write them down and go over them until you know them well.
These properties are very important when analyzing log expressions.
Let's take a look at the first property.
Log base a of one equals zero.
Because as you remember the definition of a logarithm is
the log base a of x equals y means that a to the y equals x.
We do need to make restrictions on the variables.
In this case, the base a cannot equal one and must be positive.
Let's look at an example.
Log base 4 of 1 equals 0.
This is because the base 4 when raised to the 0 power will result in 1.
So you can always rewrite a logarithm as an exponential.
The base is 4, the exponent is 0, and the result is 1.
Let's look at the next property.
Property two is log base a of a equals one.
And again we have the same restrictions on our variables.
A must be greater than zero and cannot equal one.
And if you rewrite this logarithm as an exponential,
you get a to the first power equals a.
And so you can see why it's tricky.
Take a number a, raise it to the first power you'll always get out a.
Let's look at an example.
Natural log of e equals what?
We have our natural log is log base e.
So this is saying log base e of e equals, of course it will be 1.
Since e to the first power equals e.
Okay now let's look at another property.
The next two properties are inverse properties.
The first one is a raised to log base a of M equals M.
And here we have to note again that a cannot equal one,
a and M must be positive.
Remember when you are doing a composition,
a to a power x and log base a of x are inverses of each other
meaning that they undo each other.
Whatever value goes in comes out.
In this case, M.
So for example,
if you had say 5 to log base 5 of a number like 14,
then this will give you 14
because 14 goes into the logarithm
and then you plug 5 into the x, they're undoing each other.
So 14 goes in and then 14 comes out.
The next property is also an inverse property.
It says that the log base a of a to the r power equals r
and again you can see the inverses.
Log base a of x and a to the x are inverses
so if r goes in, that's the input then that's the result.
They undo each other.
An example of this would be
the natural log of e to, say 3.2 would equal what?
Well 3.2 went in so that means, since these are inverses of each other,
then the result would be 3.2.
These are very nice to remember because
they are very easy to once you understand the inverse properties.
Let's take a look at the next couple of properties.
Let's look at the fifth property.
It says log base a of the product MN is
equal to log base a of M plus log base a of N.
I think one of the things that might help you here
if you remember the words that the log of a product equals the sum of the logs.
Here the product is M times N and that's the same thing as the sum of the logs.
And again our restrictions here are that a must be greater zero,
it can not equal one, and M and N must also be greater than zero.
Let's look at an example.
If we have log base 6 of 9 plus log base 6 of 4,
this is the right hand side.
We have the sum of logs. It will equal the log of the products.
So we have log base 6 of the product of 9 and 4, which is 36.
And now we can figure this exact value out
because what is log base 6 of 36?
What power do you have to raise 6 to in order to get 36?
The answer is 2 right.
So log base 6 of 36 equals 2.
Let's take a look at the next property.
The sixth property is log base a of a quotient M/N is
equal to log base a of M minus log base a of N.
And as with the property I did a minute ago with the log of a product,
if you say the words, I think it will help you remember the property.
This says that the log of a quotient equals the difference of logarithms.
And this example in this property, M over N is the quotient
and you can see that the log of the quotient equals the difference of the logs.
Once again, a is greater than zero, it cannot equal one,
and the arguments M and N must also be positive.
Let's look at an example of this.
If you have, say log base 7 of 14 minus log base 7 of 2,
how could you use this property and rewrite this?
You have the right hand side. You've got the difference of logs
and what does that equal to?
The difference of logs equals the log of the quotient of 14 and 2.
And now we know that 14 divided by 2 is 7
and we can find the exact value of this logarithm.
7 to what power gives you 7?
This was an earlier property that equals 1.
So I can take a difference of logs, rewrite it as a quotient,
and it is important that you remember here that the
difference of logs is the log of the quotient, not the quotient of logs.
So I think the words here will really help you to remember this property.
Okay let's look at another property.
Let's look at property number seven.
Property seven says log base a of M to a power r
equals r times log base a of M.
We still have the same restrictions on the base a that we had before.
A cannot equal one and in this case, both a and M must be positive.
R can be any real number.
An example of this would be log base 5 of x cubed is equal to what?
Well notice we have the left hand side.
R in this case is 3.
So what this property says is that we can bring the
exponent 3 down in front and write it as a product.
So the 3 comes down in front and so we have then 3 times log base 5 of x.
Let's look at the last property.
Let's look at the last property, property number eight.
It says log base a of M equals log base b of M divided by log base b of a.
This is called the change of base formula.
And as before, we have some restrictions on our variables.
Both a and b cannot equal one and a, b, and M must all be positive.
This change of base formula is very useful when
you're trying to find an approximate value for a logarithm.
Let's look at an example.
Log base 3 of 20 equals what?
Well remember logarithms and exponents.
So I'm looking for what power do I need to raise 3 to in order to get 20.
Think about that for a moment and what do you think you're going to get?
Well let's see.
3 to the second power gives me 9.
3 to the third power gives me 27.
So I realize that this is going to be somewhere between 2 and 3.
Well we can use this change of base formula and go to a different base.
Now if you know notice your calculator, look at your calculator for a moment.
It has two log buttons on it.
Its got the natural log button, ln, and its got the common log button, log.
So if you're going to use your calculator to approximate,
you want it to change either the common log base ten or the natural log base e.
We'll do both just to show you.
So using the change of base formula,
we can write this as a common log base 10 of 20.
The base here is 10, M is 20, a is 3, and we're changing to base 10.
Remember with common log, you don't see the base
and so this formula would say it would be over log base 10 again.
The base is 10 and our a value is 3.
I can also write this as a natural log of 20 divided by the natural log of 3.
Now these are all of the exact values.
If I now use my calculator and approximate this,
I'll use the last one, the natural log.
So find your natural log button, hit it.
Mine automatically gives me an open parentheses.
So you need to be careful of your calculator, whatever it does, whatever it has.
You definitely want to make sure you have natural log of 20 by itself.
And then make sure you close the parentheses or else it will continue to divide.
Close the parentheses
and then hit divided by the natural log button again and then I want 3.
Close the parentheses and then I hit equals.
And I notice that I'm going to round off at three decimal places
but you will just follow whatever the directions say in your problem.
I ended getting 2, I'm going to round to three decimal places.
I get approximately 2.727.
And notice before we started this problem we knew
it was going to be somewhere between 2 and 3.
So this answer sounds reasonable.
So this property is very useful when
you're trying to find an approximate value of a logarithm.
Now the next examples we'll see are common student errors
that students make a lot and
these are the ones you want to be very careful about.
A lot of students make errors and
you need to be careful about this.
Let me point out a couple of errors that students make most often.
The first one is that log base 4 of a sum.
Students tend to want to distribute the log through there and
write this as log base 4 of x plus log base 4 of 1.
Well this is not equal.
There is no rule for the log of a sum.
Remember properties five and six, there is a rule for a log of a product.
There is a rule for the log a quotient
but there is no rule for the log of a sum or a difference.
So don't be tempted to distribute through a sum like this.
The next one is with a difference.
For example, if you had the natural log of 2x minus 1.
That's the log of a difference.
There is no rule for the log of a difference.
That is not equal to the natural log of 2x minus the natural log of 1.
In fact, if you stop and think about it, what is the sum of logs
that we've talked about earlier?
The sum of logs is the log of a product not the log of a sum.
So you need to be very careful. Remember the words will help you.
The sum of logs is the log of the product.
The difference of logs is the log of a quotient.
So there's no rules for the log of a sum or the log of a difference.
The next error students make is they would be tempted to bring this 3 down.
It's very tempting because you remember there is
a property that says you can bring it down in front.
But that's only when the exponent applies to the argument here,
not to the whole logarithm.
This is not equal to 3 times log base 5 of x.
So don't bring down the 3 there in front when it applies to the whole logarithm.
It only comes down in front when it's applied to the argument.
Another error I see is when students have something like this.
Again, I see that the 3 is applied to the argument but
you cannot bring the 3 down in front of the whole thing
because this 3 only applies to the x.
It doesn't apply to the 6.
So you can't bring it down in front of the whole thing.
So be careful. Make note of these common student errors
and go over all those properties so you know them well.
Next thing we're going to do is look at using
properties to expand and condense log expressions.
We will also use properties of logarithms to expand and condense log expressions out.
Let's take a look at this next example.
We want to write as a sum and/or difference of logarithms.
Express powers as factors.
You see log of x cubed times square root of x minus 1
over x plus 2 to the fourth power.
The base here is 10 because you don't see it
and remember this is the common log.
When we look at this, we have to figure out
what's the big picture and have to expand it out.
The big picture here is the log of a quotient
and I remember that property that says
the log of a quotient equals a difference of logs.
So I can write this as log of the product
of these two minus the log of the denominator.
So I've got the quotient of everything in the numerator minus the denominator.
The log of a quotient equals the difference of logs.
Now within this, I see a product.
I see x cubed times the square root of x minus 1.
So I have another property that says
the log of a product equals the sum of logs.
So I can expand this out as log of x cubed
plus the square root of x minus 1
minus log of x plus 2 to the fourth power.
And we're not quite finished.
We have a sum or difference but the last step says express powers as factors.
So there is a property that says we can bring exponents down
but before we do that,
we're going to rewrite the square root of x minus 1.
Do you remember how to write the square root of x minus 1 with no radical symbol?
That's right. You write it to the 1/2 power.
So this would be log of x cubed plus
the log of x minus 1 to the 1/2 minus the log of x plus 2 to the fourth.
And then our final step is to express the powers as factors using one of our properties.
It says we can bring the exponent down in front.
So our final answer will be 3 log of x plus 1/2 log of x minus 1
minus 4 times the log of x plus 2.
So here we used three properties.
We used the log of a quotient. We used the log of a product
and we used the property that says you can bring the exponent down in front.
So we expanded this out.
The last example that we are going to do is the reverse.
We're going to write it as a single logarithm.
In this last example, we will use properties of logarithms
to condense a logarithm into a single logarithm.
Let's take a look at this example.
We have log base 3 of x squared minus 36 minus 4 log base of 3 of x plus 6
and we want to write this as a single logarithm.
You might be very tempted.
You see the difference of logs to write as a quotient right now
but you can't because of this 4.
That property, remember the difference of logs.
First of all, the bases have to be the same and they are.
The coefficients in front of the logs have to be a 1.
So we can't write it as a quotient yet.
So we have to think about what we can do with this 4.
Well remember that property that says you can bring the exponent down.
Well you can also use that property to bring that exponent up as a power.
So we can write this as log base 3 of x squared minus 36
minus log base 3 of x plus 6 to the fourth power.
So now we have the difference of logs.
It is very important the coefficients are 1.
So I have the difference of logs and I remember that property
that the difference of logs is the log of a quotient.
So now we can write this as the log base 3 of x squared minus 36
over x plus 6 to the fourth power.
So the log, the difference of logs is the log of the quotient
and notice it says to simplify.
Can we simplify this expression?
Well I notice in the numerator I have x squared minus 36
and the denominator I have x plus 6 raised to the fourth power.
Well I remember that this can be factored as the difference of squares
so we can be factored as x minus 6 x plus 6
and the denominator is x plus 6 to the fourth power.
So can we simplify this? Yes we can.
We have a factor of x plus 6 both in the numerator and the denominator.
So we can finally reduce this and simplify to log base 3 of x minus 6
over x plus 6 to the third power.
So now we've written this expanded out form and condensed it into a single logarithm
and you will need to do this when you're solving log equations later on.
You need to go over all of these properties that we've gone through very carefully
and make sure you know them very well.
They are very important when analyzing log expressions and solving log equations.
So practice all of the examples, go through them all, and then
work similar type problems on your own.