Does anyone know how to express log √(x^4 / yz) in terms of sums and differences of logarithms?

Question

Does anyone know how to express log √(x^4 / yz) in terms of sums and differences of logarithms

Answer 1

Yes. Remember your log rules:
log (xy) = log x + log y
log (x/y) = log x - log y
log (x^a) = a log x

Also, √x = x^(1/2), so we get
log √x = log (x^(1/2)) = (1/2) log x

So, log √(x^4 / yz)
= log ((x^4 / yz)^(1/2))
= (1/2) log (x^4 / yz)
= (1/2) (log (x^4) - log (yz))
= (1/2) (4 log x - (log y + log z))
= 2 log x - (1/2) log y - (1/2) log z.

You could also choose to evaluate the square root before the log, and write everything as a multiplication: you'd have
log √(x^4 / yz)
= log ((x^4.y^(-1).z^(-1))^(1/2))
= log (x^2.y^(-1/2).z^(-1/2))
= log (x^2) + log (y^(-1/2)) + log (z^(-1/2))
= 2 log x - (1/2) log y - (1/2) log z
as before.

Answer 2

Education is good, because it teaches you how to learn, how to discipline your mind. Knowledge for a Christian is knowing the word, knowing your God, His nature, His will, His outlook on things. Wisdom is looking at things from God's perspective and putting them into action. Knowledge is great, but too much of it can puff you up if you don't counteract it with fruit of the Spirit. Ever know a person who's very knowledgeable, but has no temperance and no love? Their knowledge is almost useless in the long run. But wisdom? You find that and your soul shall live, because when you find wisdom, you find the way God Himself looks at a situation. Proverbs 3: By wisdom the Lord laid the earth’s foundations, by understanding he set the heavens in place; by his knowledge the watery depths were divided, and the clouds let drop the dew.

Answer 3

log of a product is the sum of the logs; log of a fraction is log of numerator minus log of denominator. log of a power is the power times the log.

log√[x^4/yz] = 0.5*log(x^4/yz) = 0.5*[log(x^4) - log(yz)] = 0.5*[4*log(x) - log(y) - log(z)] = 2*log(x) - 0.5*log(y) - 0.5*log(z)