## anonymous 2 years ago "Write each logarithmic expression as a single logarithm." 1/2(logx4+logxy)-3logxz

1. anonymous

Hints: Recall that a$$\log b=\log(b^a)$$, $$\log(ab) = \log a + \log b$$ and $$\log\left(\dfrac{a}{b}\right) = \log a - \log b$$. How would you go about writing that expression as a single logarithm? :-)

2. anonymous

I don't know. That is why I asked for help.

3. anonymous

We first apply the property $$\log(ab) = \log a + \log b$$ to see that \large \begin{aligned}\dfrac{1}{2}\left(\log(x^4) +\log (xy)\right)-3\log(xz) &=\dfrac{1}{2}\log(x^4\cdot xy) - 3\log(xz)\\ &=\dfrac{1}{2}\log (x^5y)-3\log (xz)\end{aligned} We now apply the property $$a\log b = \log (b^a)$$ to get\large \begin{aligned}\frac{1}{2}\log(x^5y) - 3\log(xz) &= \log((x^5y)^{1/2})-\log((xz)^3)\\ &= \log (x^{5/2}y^{1/2}) - \log (x^3z^3)\end{aligned}Lastly, we now apply the property $$\log\left(\dfrac{a}{b}\right) = \log a-\log b$$ to see that \large\begin{aligned} \log(x^{5/2}y^{1/2}) - \log(x^3z^3) &= \log\left(\frac{x^{5/2}y^{1/2}}{x^3z^3}\right)\\ &= \log\left(\frac{y^{1/2}}{x^{1/2}z^3}\right)\end{aligned}Does this make sense? :-)