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Express the following in the form $a\log x + b\log y + c \log z$: $$\log\sqrt[4]{\frac{x^9}{y^4z^3}}$$

I'm struggling to find a way to approach the question. Any ideas on how I would answer or even start this problem?

My attempt:

= 1/4(logx^9 - logy^4 + logz^3) = 9/4(log(x)) +log(y) + 3/4(log(z))

I got it, I think the answer is

9/4(logx) - 3/4(logz) - logy

Anyways, thanks to anyone who tried to help.

Carstairs
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  • Um... do you know any log identities? $\log mn = \log m + \log n$ and and you know $\log (m)^k = k\log m$ and $\sqrt[k]{m} = m^{\frac 1k}$ and then $\frac 1m = m^{-1}$. So this should be exceedingly easy. There is utterly no trick or anything unusual being tossed. – fleablood Sep 29 '18 at 00:26
  • I attempted it, but I got 9/4log - logy + 3/4logz which I was told is wrong, and I don't know where along my solution I went off. – Carstairs Sep 29 '18 at 00:29
  • @Carstairs: You should include your solution as part of the question. If you show your work, we may be able to help identify where you went wrong. – Blue Sep 29 '18 at 00:32
  • $a - (b + c) \ne a - b + c$. Instead $a - (b+c) = a - b -c$. – fleablood Sep 29 '18 at 00:33
  • "which I was told is wrong" Hmmm, they should have told you it was wrong but that it was close and you just made a math error. – fleablood Sep 29 '18 at 00:39

2 Answers2

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$\log \sqrt[4] M = \log M^{\frac 14} = \frac 14 \log M$.

$\log \frac mn = \log m - \log n$.

$\log k^9 = 9\log k$

And $\log ab = \log a + \log b$.

So $\log \sqrt[4]{\frac {x^9}{x^4y^3}} = ......$?

There is no trick and there is no curveball.

You just do it.

fleablood
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Sure. Use the rules for logs: $$\begin{align} \log x^n&=n\cdot \log x \\[8pt] \log ab&=\log a+\log b\\[8pt] \log \frac ab&=\log a-\log b \end{align}$$

Blue
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