Another logarithm question.... thanks for your patience.?

Question

No, this is not homework. It is a test review, and I am desperately trying to figure out HOW to do these so I will do alright on the actual test.

Thanks to all who help. (By the way, I'm in college Pre-calculus and am struggling because I haven't had a math class since high school 15 years ago.) Put this with the demanding hours of my job, and I'm really struggling to hold onto my avg.
You don't know how much I appreciate the help that you provide.




3 ln(x+1) - 2 ln y + ln(x^2 + 4)

condense

becomes [ ln (x+1)^3 (x^2+4) all over y^2]

My question is, why is the last piece - the (x^2+4) also over the y^2. I would think it would be separate. Can anyone explain that part?

Answer 1

3[ln(x + 1)] - 2[ln y] + ln(x² + 4)
ln(x + 1)³ - ln y² + ln(x² + 4)
ln[(x + 1)³ / y²] + ln(x² + 4)
ln[(x + 1)³ / y² * (x² + 4)]
So, it sounds like you got this far.
Keep in mind that multiplication and division are commutative. It can be rewritten as:
ln[(x + 1)³(x² + 4) / y²]

Answer 2

Okay, so you seem to accept that a*ln x + b*ln y = ln(x^a * y^b), and that a*ln x - b*ln y = ln(x^a / y^b).
The problem you are struggling with just does this twice: first there is a subtraction problem, then some addition. It might be that what is throwing you is simply the fact that A*B/C = (A*B)/C = (A/C)*B, or in this case that ((x+1)^3)(x^2+4)/(y^2)=
((x+1)^3)/(y^2))(x^2+4).

Try it with numbers:

(8*6)/4 = (8/4)*6 because 48/4=12 and 2*6=12.

Answer 3

The reason
(x^2 + 4)
is in the numerator is because it's positive. Remember, a logrithm is an exponent. So the argument of
+ ln (x^2 + 4)
will be on top while, the argument of
- 2 ln y
will be a denominator once condensed. You obviously know how to condense but seem to have a problem trying to figure whether the argument is on top or bottom, basically if it's negative after you simplify it goes down and positive up (after you simplify att signs of course).

Answer 4

ln a +ln b = ln ab
ln a - ln b = ln (a/b)
So 3 ln(x+1) - 2 ln y + ln(x^2 + 4)
= ln(x+1)^3 +ln(x^2+4) -lny^2
= ln[(x+1)^3)(x^2+4)/y^2]

Answer 5

So far you have:
[(x+1)^3 / y^2] * (x^2 + 4)

If you look at that in simpler terms...

(a/b) * c
you would simplify that to be
(a/b) * (c/1) = (a*c)/(b *1) = ac/b

Answer 6

log a + log b = log (ab)
loga - logb = log( a/b)
a log b = log ( b^a)

the -2lny = +log (1/y^2)

note that a * (b/c) = (a*b)/c if that is what puzzled you. its just a stylistic issue how you write it