Question

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

Answer 1

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? :-)

Answer 2

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

Answer 3

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? :-)