The topic here is writing as a single logarithm.
So let do more examples.
So let do example four.
Write as a single logarithm the expression
five log of x plus one, plus three log of
x minus two, minus fifteen log of two x plus
eight.
We always use when we go backward.
In other words, when we try to write as a
single log we always see if we can use three
first, if we can use property three first.
Can we?
Yes.
What does property three tell us?
Tell us that five can be moved here that three
can move here and the fifteen can move here.
So this is going to be equal to log of x plus
seven to the fifth, plus log of x minus two
three, minus log of two x plus eight fifteen.
So, this is what three.
Now after that we always do, we go from left
to right.
In other words, we need to do, to rewrite
these two logs; the sum of these two logs
as one log.
See since it's a sum here then we have to
use property number one right.
So property number one says that will be equal
to log of x plus seven to the fifth times
x minus two cubed and don't forget minus log
of two x plus eight to the fifteen.
Now we have two logs we're down from three
logs to two log now we need to go to one log.
So since we have a difference here, right
there so we need to use number two.
So number two tell us we're going to get division.
So this is log of x plus seven to the fifth,
x minus two cubed , everything over two x
plus eight fifteen, and we're done; see we
got one log.
Now for example five again same thing we're
always start with number three if we can.
Number three tell us this exponent move here,
so let's do that this is going to be equal
to log of x plus two to one third, minus log
of x minus 1 squared, minus log of x cubed
plus x to the fifth.
Now, we're , like I said we're always start
from left to right . So we're going to write
these two logs as one.
Since we have a difference we 're need to
use number two, right . Number two again,
number two says we're going to get a division.
So log of x plus two one third over x minus
one squared and then minus log of x cubed
plus x to the fifth.
Now we're down to two logs, we need to go
back all the way, we need to go all the way
to one log.
Since this is a difference again difference
means we are still going to use number two
one more time.
Number two says this is equal to log of x
plus two to the third; we can we can rewrite
as cube root, right.
Cube root of x plus two, so we have that x
minus one square divide well let's just write
it this way, divide by x cubed plus x fifth,
right.
This doesn't have space on my hands division
the old traditional way.
Now, what does that mean division?
Division means multiplication by the inverse,
so this is going to be equal log of cube root
of x plus two over x minus one squared times
one over x cubed plus x to the fifth.
This looks better, right.
And then of cause we can rewrite it as log
of anything times one is itself plus two over
x minus one square x cubed plus x to the fifth.
And that's set for this example.
