Iin this lesson
I just wanted to walk through some guidelines for condensing logarithms. Condensing logarithms is important in order to solve logarithmic
equations which we're going to be seeing in the next section. So it's important that we still have a good understanding
of those log rules that we saw earlier in this section. If you have any questions on those log rules
I would say to either go back and watch Professor Messenger's video or the other lesson on
expanding logarithms.
When we are condensing logarithms think of our order of operations
In terms of we want to work inside parentheses first or parentheses brackets braces whatever
we may have. In this case we can see
that we have a square bracket with the 1/3 being multiplied on the outside and then within the parentheses of each of these natural logs
we have
an x plus 6 and then an x squared minus 25. Now
it's important to understand that the x plus 6 is attached to that natural logs.
We're taking the natural log of x plus 6 that x plus 6 is just going to stay there.
Same thing with the x squared minus 25 it is within the natural log since we're taking the natural log x squared minus 25.
So those are going to stay. So when I talk about working inside parentheses or brackets,
we're looking at having multiple logs within a set of parentheses or brackets which we have right now
we have three logs within one braces.
So here sort of working backwards from what we did with our expanding our logarithms
we first want to use the power rule to bring up any numbers that are in front of individual logs.
Okay, so if I look at my first log again
I have five natural log of x plus six. That five
I'm going to want to bring up as an exponent. So this one-third is going to stay.
My bracket is going to stay,  but the five is going to come up so that we have the natural log of x plus six
raised to the fifth power.
Looking at my next log. I have the natural log of x there's no number in front of natural log
so there's nothing to do. And then our last one looking at the natural log of x squared minus 25
there's no number in front so that also is going to stay the same so these last two logs
we don't
touch. The only thing that we were able to do in our first step was to bring up that five that was in front of
that natural log.
Our next step is sort of a two-part step essentially we want to do is we need to look
from left to right so again
I'm still working inside of those square brackets
and we're going to use either our quotient rule to condense two logs with subtraction into one log with division or
we're going to use our product rule to condense two logs with addition into one log with multiplication.
So looking inside of our brackets moving left to right
I have two logs first two logs that I see are separated by subtraction.
Since those two logs are separated by subtraction, we're going to condense just those two logs into one log with division.
That means we're going to have this one third
still on the outside. Then again remember
it's becoming two two logs turn into one log, so we have one natural log
and then we're going to make this into our quotient rule.
So our first log that x plus six to the fifth will be our numerator, and
then our denominator will be what we got from our second log which would just be x.
So those two logs disappear and using our quotient rule turn into one log which means I still have this plus
natural log of x squared
minus 25.
So when we're looking at this again
it's important to understand that we started off with the natural log of x plus 6 to the fifth minus the natural log of x.
When I condensed it, there's only
one log, It goes completely on the outside, and then I use the parentheses to show the numerators x plus 6 to the fifth all
over x.
So that was our first step now again like I said the second step kind of has a few parts to it because we do
want to move left to right . First
we were able to use the quotient rule,
but I can continue to use either the quotient rule of the product rule.
And you want to repeat this as many times as necessary
either using your quotient rule your product rule one at a time moving left to right.
But if I look at what I have right now
still inside of our brackets, I had two logs that are
separated by an addition. Well two logs separated by an addition. I can use our product rule and we can condense it with multiplication.
So
what we have here then would be the one third still on the outside
natural log of x plus six to the fifth
all over x and then times that x squared
minus
25.
So what happened was we got rid of that second log now this isn't exactly the prettiest
answer that we have we do want to make this into one fraction.
So remember when we're multiplying we have a fraction in a whole number you can put that whole number that x squared minus 25
considering to be that whole number part over one and when we multiply it it actually goes into the numerator.
So whenever you have a fraction
and you're going to be using your product rule, really what's going to happen you can shorthand
this is to put in this case that x squared minus 25 into the numerator. So
that way we have 1/3
natural log of
x plus 6
to the fifth and then that x squared minus 25 will go in the numerator all
over x.
So that's what we have right now. For
a third step we want to bring up any remaining numbers that are in front of natural logs or logs using our power rule.
So again, we started off using our power rule
and now we're going to be closing out with using our power rule . If you look at how I
rewrote our answer from the previous slide what I did was I got rid of the brackets that were around the natural log.
Which is perfectly fine since we had completely condensed what was inside.
This way you can see that that 1/3 is being multiplied by the natural log which means that we can
bring it up. And
remember again to save yourself time you can say that you have the natural log of
the whole inside raised to the one-third but a one-third exponent does mean that's going to be a cube roof.
So we're keeping this as the natural log of
the cube root of
our x
plus six to the fifth
x squared minus 25
all
over x. And
that would be our condensed form so again remember that the natural log is still on the outside of
that root or whatever the exponent happens to be. Our very last step would be to try to simplify and reduce if possible.
The only thing that I could do here would be to
factor
that x squared minus 25, so I would have x plus 6 to the fifth and
then x squared minus 25 will become x plus 5 x minus 5,
all over x. So although I can factor it really what we have here
cannot be simplified any more so either of these two answers would be fine .Again the important piece in terms of bringing up that last
exponent using the power rule is the natural log has to be on the outside.
