Simple question about writing a log-how does the root go turn to divison?

[singleton]

Simple question about writing a log--how does the root go turn to divison?

I understand the product, quotient and power laws of logarithms. But this piece in the book has me stumped. It says how they used logarithms to rewrite calculations and make them easier to figure out. Fair enough. But I'm not sure how part of it happens! (and the book sucks at expanding/explanation!)

(31.7^5 x 0.64 / 171.8)^1/3

becomes

1/3{5log31.7 + log0.64 - log171.8}

I understand what is inside of the braces, but how can the exponent come down in front like that

I can understand {5log31.7 + log0.64 - log171.8}^1/3 but not what is stated!

 

[courtrigrad]

Hi

Do you know your log rules?

log (AB) = log A + log B

log (A / B) = log A - log B

log (a^c) = c*loga

Going to your problem we first work inside the brackets.

31.7^5 x 0.64 / 171.8 is what we will work with first.

So take the log of the numerator. It is a product. We have

log (31.7^5 * 0.64) / log (171.8). Use your log rules.

log (31.7^5) = 5*log(31.7)

= 5*log(31.7) + log 0.64 - log 171.8 (By our rules, try to figure it out)

Now we work with the power.

(5*log(31.7) + log 0.64 - log 171.8) ^ 1/3

= 1/3 (5* log(31.7) + log 0.64 - log 171.8)

Hence our problem is solved.

 

[singleton]

Sorry I should have stated how far I understand.

I understand the log laws and I can get to
(5*log(31.7) + log 0.64 - log 171.8) ^ 1/3

but I couldn't get the 1/3 in front

OHHHHH

frig
I wasn't even thinking about the power law :/
I was thinking in terms of solving what is inside the brackets (5.076)^1/3

I wasn't thinking of terms of the brackets having the log laws apply to them

I'm so stupid

Edit: What I meant to say was that, I wasn't even thinking of what is inside the braces in terms of a log, rather I saw the "final solution" in my head and couldn't understand how it to the exponent 1/3 worked out that way. It makes better sense now.

Thanks ;)