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

Question

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.

Answer 1

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

Answer 2

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

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

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

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.