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

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ChristopherToni
 11 months ago
Best ResponseYou've already chosen the best response.1Hints: 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? :)

Emily778
 11 months ago
Best ResponseYou've already chosen the best response.0I don't know. That is why I asked for help.

ChristopherToni
 11 months ago
Best ResponseYou've already chosen the best response.1We 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? :)
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