Problem 1
Jika $\displaystyle \log_{xy}( y) =\frac{1}{4}$, maka nilai dari $\displaystyle \log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right)$ adalah ...
| a) $\displaystyle \frac{11}{24}$ |
| b) $\displaystyle \frac{1}{2}$ |
| c) $\displaystyle \frac{7}{12}$ |
| d) $\displaystyle 1$ |
| e) $\displaystyle \frac{9}{16}$ |
Modifikasi $\displaystyle \log_{xy}( y) =\frac{1}{4}$,
\begin{align*}
\log_{xy}( y) & =\frac{1}{4}\\
\log_{y}( xy) & =4\\
\log_{y}( x) +\log_{y}( y) & =4\\
\log_{y}( x) & =3\\
\log_{x}( y) & =\frac{1}{3}
\end{align*}
Modifikasi $\displaystyle \log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right)$,
\begin{align*}
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\log_{xy}\left(\frac{x^{\frac{2}{3}} y^{\frac{4}{3}}}{y^{\frac{3}{2}}}\right)\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\log_{xy}\left( x^{\frac{2}{3}} y^{\frac{4}{3} -\frac{3}{2}}\right)\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\log_{xy}\left( x^{\frac{2}{3}} y^{-\frac{1}{6}}\right)\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\log_{xy}\left( x^{\frac{2}{3}}\right) -\log_{xy}\left( y^{\frac{1}{6}}\right)\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{2}{3}(\log_{xy}( x)) -\frac{1}{6}(\log_{xy}( y))\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{2}{3}\left(\frac{1}{\log_{x}( xy)}\right) -\frac{1}{6}\left(\frac{1}{4}\right)\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{2}{3}\left(\frac{1}{\log_{x}( x) +\log_{x}( y)}\right) -\frac{1}{24}\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{2}{3}\left(\frac{1}{1+\frac{1}{3}}\right) -\frac{1}{24}\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{2}{3}\left(\frac{3}{4}\right) -\frac{1}{24}\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{1}{2} -\frac{1}{24}\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{11}{24}
\end{align*}
Jawaban: a
Problem 2
Jika $\displaystyle x_{1}$ dan $\displaystyle x_{2}$ adalah akar-akar persamaan dari $\displaystyle 3x^{2} -2x+a=0$ dan memenuhi $\displaystyle \frac{1}{x_{1}} +\frac{1}{x_{2}} < 1$, maka pernyataan yang paling benar adalah ...
| a) $\displaystyle a >2$ |
| b) $\displaystyle a< 0$ atau $\displaystyle a >2$ |
| c) $\displaystyle 0< a< 2$ |
| d) $\displaystyle a< 0$ |
| e) $\displaystyle a< 2$ |
Modifikasi $\displaystyle \log_{xy}( y) =\frac{1}{4}$,
\begin{align*}
\log_{xy}( y) & =\frac{1}{4}\\
\log_{y}( xy) & =4\\
\log_{y}( x) +\log_{y}( y) & =4\\
\log_{y}( x) & =3\\
\log_{x}( y) & =\frac{1}{3}
\end{align*}
Modifikasi $\displaystyle \log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right)$,
\begin{align*}
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\log_{xy}\left(\frac{x^{\frac{2}{3}} y^{\frac{4}{3}}}{y^{\frac{3}{2}}}\right)\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\log_{xy}\left( x^{\frac{2}{3}} y^{\frac{4}{3} -\frac{3}{2}}\right)\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\log_{xy}\left( x^{\frac{2}{3}} y^{-\frac{1}{6}}\right)\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\log_{xy}\left( x^{\frac{2}{3}}\right) -\log_{xy}\left( y^{\frac{1}{6}}\right)\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{2}{3}(\log_{xy}( x)) -\frac{1}{6}(\log_{xy}( y))\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{2}{3}\left(\frac{1}{\log_{x}( xy)}\right) -\frac{1}{6}\left(\frac{1}{4}\right)\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{2}{3}\left(\frac{1}{\log_{x}( x) +\log_{x}( y)}\right) -\frac{1}{24}\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{2}{3}\left(\frac{1}{1+\frac{1}{3}}\right) -\frac{1}{24}\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{2}{3}\left(\frac{3}{4}\right) -\frac{1}{24}\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{1}{2} -\frac{1}{24}\\
\log_{xy}\left(\frac{\sqrt[3]{x^{2} y^{4}}}{\sqrt{y^{3}}}\right) & =\frac{11}{24}
\end{align*}
Jawaban: a
