Professor Dave again, let’s learn more about logs.
Now let’s learn a few properties of logarithms.
First, if the base of the log and the number that the log operates on are the same, this
will always be one, because anything to the first power equals itself.
If instead, we are taking the log of one, this will always be zero, regardless of the
base, because any base must be raised to the zero power to get one.
Another important property is that log base B of B to the X is always X, since this can
be expressed as B to the X equals B to the X.
So when you see the log of an exponential term where the bases match, it all goes away
and we are just left with the exponent.
Similarly, B to the log base B of X also equals X, because this log represents the number
B must be raised to in order to get X, so if B is raised to that number, we must get X.
Now let’s learn some properties that will allow us to do operations with logarithmic terms.
These properties will correspond to properties of exponents that we already know.
For example, we know that X to the A times X to the B equals X to the A plus B, which
allows us to combine terms, or split one term up into two.
Because logarithms are essentially exponents, then it should follow that for any base, including
base ten, which we will assume for the time being, log of AB equals log of A plus log
of B. That means we can take something like log of ten X and turn it into log of ten plus
log of X. Log of ten is one, so this becomes log of X plus one.
We also know that X to the A over X to the B equals X to the A minus B, which again can
be used to expand or condense certain expressions.
Therefore it is the case that log of A over B equals log of A minus log of B.
This means that with something like log of three over X, we could express it as log of
three minus log of X.
Next, we can remember that X to the A to the B equals X to the A times B. The corresponding
property for logs tells us that log of X to the A equals A log X.
So if we have log X squared, that is the same as two log X.
Let’s quickly mention that this only applies if the exponent is part of the term that the
logarithm is operating on.
If we had log of X quantity squared, that just means log of X times log of X, just as
with anything else that we could square.
This is not related to the property we are describing.
When this exponent is in here, on this X term, it means that we are taking the log of X squared,
and it is in this case that we can pull the two over to the front and get two log X.
So these are the three ways that we can expand logarithmic expressions.
They are the product rule, the quotient rule, and the power rule.
Now that we know these rules, let’s practice a bit.
Try log base four of the square root of X over sixteen Y squared.
The first thing we notice is that we have a quotient, so let’s break this into log
of root X minus log of sixteen Y squared.
This second term has a product, so we can expand again.
But we have to be careful about our signs.
To be safe, let’s put this term in parentheses first.
After expanding we get log sixteen and log Y squared.
Now to get rid of the parentheses, we have to distribute this negative sign, so both
of these terms become negative, and we are now done with that step.
We can actually simplify further.
Starting on the left, let’s rewrite root X as X to the one half.
We know that we can pull exponents to the front, so this becomes one half log X.
Then, log base four of sixteen is two, because four squared is sixteen, so this becomes minus two.
Lastly, we can pull the exponent to the front on this last term to get minus two log Y,
and there is our expanded expression.
Just for fun, let’s take some other expression and see if we can get it to condense into
a single log.
How about two plus two log X.
If we want to combine these into one term, we need logs on each term, so two has to become a log.
The other log is implied to be base ten, so how can we express the number two as a log
base ten?
Well ten squared is a hundred, so log of a hundred is the same as the number two.
That gives us log of a hundred plus two log X.
Now just the way we can bring an exponent to the front, we can push a coefficient up
over this term, so two log X can be log X squared.
And now we have a sum of logs with the same base, so we can combine them to get a product.
And log one hundred X squared is what we get.
I think that’s plenty of practice, so let’s check comprehension.