[MUSIC] Let's work with the Properties of Logarithms.
For example, given that log x=3 and log y= -2, let's find the log of x to the
5th, y cubed. And the log of x squared divided by
square root of y. And we'll be using the following
properties of logs to help us. Where here a is any positive base not
equal to 1, M and N are positive numbers and P is any number.
So let's begin with the log of x to the 5th, y cubed.
Cubed. By this property down here, the logarithm
of a product is the sum of the logs, in other words this is equal to log of x ^ 5
+ log of y ^ 3. And by this last property down here, the
logarithm of a power, we can take this power and bring it down in front of the
logs. So let's do that to both of these
logarithm expressions here. We can the 5 down in front here, as well
as the 3 down in front here. Which gives us, 5 x log of x plus 3 x log
of Y. Now we're given that log x = 3, which we
can plug in here, as well as log y = -2, which we can plug in here.
Therefore, this is equal to 5*3, + 3*-2 or 15-6, which is 9.
Alright, and what about this last log we are asked to find? The log of x^2 divided
by the square root of y. Well, by the second property over here,
the logarithm of the quotient is equal to the difference in the logarithms.
That is, this is equal to log x ^ 2 - log of the square root of y.
Which is equal to log x ^ 2 and then minus log, and we can rewrite square root
of Y as Y of the 1/2 power. We wanted to write both of these
expressions with power's, so that we can bring these powers down in front of the
log, by this log rhythm with a power property down here.
That is, this is equal to 2 times log of X, minus 1 half, times log of 1.
Again we can use the fact that log X is equal to 3, and that log Y is equal to
negative 2. Which gives us 2 * 3 - 1/2 * -2 or 6 + 1
which = 7, which is our answer.
Alright, let's look at another example. [SOUND] Let's compute log base 2 of 24 -
log base 2 of 3. Now do we know a power that we can raise
2 to to get 24 or a power we can raise 2 to, To get 3.
We don't, so we could use a change of base formula
on each of these logs separately. But it's going to much easier if we use
these properties of logs. If we look here at this middle property,
we can use it in the reverse direction. In other words, we're starting with this,
and then we're going to condense it using this property.
That is, this is = log base 2 of 24 / 3, which = log base 2 of 8, which is = 3.
Since inverting to exponential form, we know that 2 cubed is equal to 8.
So this would be our answer. And this is how we work with properties
of logs. Thank you, and we'll see you next time.
[MUSIC]