Use the proportion of logarithms to expand the expression...

[DemiGod]

Use the proportion of logarithms to expand the expression...

ln [ (4x^5 - x - 1) * SqRt(x - 7) / (x^2 + 1 )^3 ]

and

ln [ (2x - 1) / SqRt(3x + 1) * (x^3 - 7)^9 ]

.... ? No idea where to start.

 

[galactus]

No idea?. If you're studying logs, you should know of the log laws.

\(\displaystyle ln(ab)=ln(a)+ln(b)\)

\(\displaystyle ln(a/b)=ln(a)-ln(b)\)

\(\displaystyle ln(a^{b})=bln(a)\)

Give it a whirl using the above laws.

 

[DemiGod]

I wasn't sure whether I should work from the inside out or seperate the logs first.

So basically just split it into (ln1 + ln2) - (ln3) and then take the log of all of it which would basically remove the ln's and then simplify?

 

[Mrspi]

Use the proportion of logarithms to expand the expression...

ln [ (4x^5 - x - 1) * SqRt(x - 7) / (x^2 + 1 )^3 ]

and

ln [ (2x - 1) / SqRt(3x + 1) * (x^3 - 7)^9 ]

.... ? No idea where to start.

It has already been suggested that you use the rules of logs...AND those rules were nicely stated for you.

ln [ (4x^2 - 4x - 1) * sqrt(x - 7) / (x^2 + 1)^3]

using the rule that log (ab) = log a + log b, we have this (the same rules apply to natural logs...so ln (ab) = ln a + ln b......)

ln (4x^2 - 4x - 1) + ln [sqrt(x - 7) / (x^2 + 1)^3]

Next, use the rule which says that log(a/b) = log a - log b (or ln (a / b) = ln a - ln b:

ln (4x^2 - 4x - 1) + ln (sqrt(x - 7)) - ln [(x^2 + 1)^3]

write sqrt(x - 7) as (x - 7)^(1/2), and use the rule of logs which says that log a^n = n log a (or with natural logs, ln a^n = n ln a):

ln (4x^2 - 4x - 1) + ln [ (x - 7)^(1/2)] - ln [(x^2 + 1)^3]

ln (4x^2 - 4x - 1) + (1/2)*ln(x - 7) - 3 ln (x^2 + 1)

Unless I've missed something here, I think that's it.

 

[DemiGod]

Wow I thought I had to actually simplify the whole expression after splitting the logs.

I find myself trying to do a lot more work than necessary when the answer is usually so simple.

I thank you guys a great deal for taking the time to give me this help. It allows me to see these problems in a way that I can take into other more complex problems where I wouldn't normally expect such a simple solution. :?

Appreciate it.