So we're trying to write each expression as
a sum, difference, or product of logarithms
and simplify if possible.
In other words, we want to expand.
And of course, we need to use this logarithmic
properties, 5, 6, and 7.
So for example 1, we have log of x 4 times
3x minus 7 to the eighth.
So since the inside is a product, so we're
going to use number 5.
So we're using number 5.
Let's do number 5.
So it's going to be log of, x 4 is the first
factor, plus log of the second factor, which
is 3x minus 7 to the eighth.
Now for each one of them, we can use number
7 right here.
Why?
Because we have an exponent.
So using number 7, we get 4 will be moving
this way, so this is going to be 4 log of
x plus, and then the eighth is going to be
moving here, that's what number 7 says, 8
log of 3x minus 7.
So that's it for number 1.
For number 2, I mean, first we need to rewrite
the cube root.
So the top is actually the same thing as x
plus 1 to the one-third.
That's what cube root means.
And then, the bottom, we're going to leave
it the same way, x to the fifth.
Now, we need to use number 6 because we have
a division.
See, when we have division, we use number
6, x over y.
So this is going to be equal, I'm using number
6, log of the numerator minus log of the denominator,
right?
So log of the top minus log of the bottom.
And for each one, since we have an exponent
here, for each one, we're going to move the
exponent here.
We're going to move this exponent here.
And that's using number 7.
So by using number 7, we get 1 over 3 log
of x plus 1 minus 5 log of x.
Now for example 3, of course, since we have
division, we're going to start with number
6.
So using number 6, we need, we're going to
get log of the top, which is x plus 7 to the
fifth, 2x minus 1, minus log of the bottom,
which is 3x plus 5 to the fourth.
And now for this one, since we have a product,
we need to use number 5.
So this is going to be equal log of the first
factor, which is x plus 7 to the 5, plus log
of the second factor, which is 2x minus 1,
minus log of 3x plus 5 to the fourth.
So this is, we used number 5 here.
Now last step, we're going to use number 7.
So again, we're going to move the exponent
here, that's what number 7 says, so this is
5 log of x plus 7 plus, since there's no exponent
here, we're not going to use anything, that's
going to be 2x minus 1, and then the 4 is
going to go here using number 7, so 4 log
of 3x plus 5.
And that's about it for this problem.
