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Dear Students,
Welcome to today’s quiz! This is your opportunity to demonstrate what you’ve learned so far, so do your best. Please keep in mind that you have a maximum of 30 minutes to complete all the questions. Make sure to manage your time wisely and answer each question thoughtfully.
Good luck!
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Pertanyaan 1 dari 10
1. Pertanyaan
1 pointsIf \(\log_{}{0.04} = m\), then \(\log_{}{0.25} = \dotso\)
Benar
\(\log_{}{0.04} = m\)
\(\log_{}{\frac{4}{100}} = m\)
\(\log_{}{\frac{1}{25}} = m\)
\(\log_{}{1}\:-\:\log_{}{25} = m\)
\(0\:-\:\log_{}{25} = m\)
\(\log_{}{25} = -m\)
\(\log_{}{0.25} = \log_{}{\frac{25}{100}}\)
\(\log_{}{0.25} = \log_{}{25}\:-\:\log_{}{100}\)
\(\log_{}{0.25} = -m\:-\:\log_{}{10^2}\)
\(\log_{}{0.25} = -m\:-\:2\)Salah
\(\log_{}{0.04} = m\)
\(\log_{}{\frac{4}{100}} = m\)
\(\log_{}{\frac{1}{25}} = m\)
\(\log_{}{1}\:-\:\log_{}{25} = m\)
\(0\:-\:\log_{}{25} = m\)
\(\log_{}{25} = -m\)
\(\log_{}{0.25} = \log_{}{\frac{25}{100}}\)
\(\log_{}{0.25} = \log_{}{25}\:-\:\log_{}{100}\)
\(\log_{}{0.25} = -m\:-\:\log_{}{10^2}\)
\(\log_{}{0.25} = -m\:-\:2\) -
Pertanyaan 2 dari 10
2. Pertanyaan
1 points\(\log_{x^2}{\sqrt{x}} + \log_{x^4}{\frac{1}{\sqrt{x}}}\:-\:\log_{x}{x^2\sqrt{x}} = \dotso\)
Benar
\(\log_{x^2}{\sqrt{x}} + \log_{x^4}{\frac{1}{\sqrt{x}}}\:-\:\log_{x}{x^2\sqrt{x}}\)
\(\log_{x^2}{x^{\frac{1}{2}}} + \log_{x^4}{x^{-\frac{1}{2}}}\:-\:\log_{x}{x^{\frac{5}{2}}}\)
\(\log_{x^2}{x^{\frac{1}{2}}} + \log_{x^4}{x^{-\frac{1}{2}}}\:-\:\log_{x}{x^{\frac{5}{2}}}\)
\(\frac{1}{4}\cdot\log_{x}{x} \:-\:\frac{1}{8}\cdot\log_{x}{x}\:-\:\frac{5}{2}\cdot\log_{x}{x}\)
\(\frac{1}{4}\:-\:\frac{1}{8}\:-\:\frac{5}{2}\)
\(\frac{2}{8}\:-\:\frac{1}{8}\:-\:\frac{20}{8}\)
\(\frac{2\:-\:1\:-\:20}{8}\)
\(\frac{-19}{8}\)
\(-2.375\)
Salah
\(\log_{x^2}{\sqrt{x}} + \log_{x^4}{\frac{1}{\sqrt{x}}}\:-\:\log_{x}{x^2\sqrt{x}}\)
\(\log_{x^2}{x^{\frac{1}{2}}} + \log_{x^4}{x^{-\frac{1}{2}}}\:-\:\log_{x}{x^{\frac{5}{2}}}\)
\(\log_{x^2}{x^{\frac{1}{2}}} + \log_{x^4}{x^{-\frac{1}{2}}}\:-\:\log_{x}{x^{\frac{5}{2}}}\)
\(\frac{1}{4}\cdot\log_{x}{x} \:-\:\frac{1}{8}\cdot\log_{x}{x}\:-\:\frac{5}{2}\cdot\log_{x}{x}\)
\(\frac{1}{4}\:-\:\frac{1}{8}\:-\:\frac{5}{2}\)
\(\frac{2}{8}\:-\:\frac{1}{8}\:-\:\frac{20}{8}\)
\(\frac{2\:-\:1\:-\:20}{8}\)
\(\frac{-19}{8}\)
\(-2.375\)
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Pertanyaan 3 dari 10
3. Pertanyaan
1 pointsIf \(\log_{3}{2} = p\) and \(\log_{5}{3} = q\), then \(\log_{18}{3.2} = \dotso\)
Benar
\(\log_{18}{3.2} = \frac{\log_{3}{3.2}}{\log_{3}{18}}\)
\(\log_{18}{3.2} = \frac{\log_{3}{\frac{32}{10}}}{\log_{3}{2\cdot 3^2}}\)
\(\log_{18}{3.2} = \frac{\log_{3}{32}\:-\:\log_{3}{10}}{\log_{3}{2}+\log_{3}{3^2}}\)
\(\log_{18}{3.2} = \frac{\log_{3}{2^5}\:-\:\log_{3}{2\cdot 5}}{\log_{3}{2}+\log_{3}{3^2}}\)
\(\log_{18}{3.2} = \frac{5\cdot\log_{3}{2}\:-\:\log_{3}{2}\:-\:\log_{3}{5}}{\log_{3}{2} + 2\cdot\log_{3}{3}}\)
\(\log_{18}{3.2} = \frac{5p\:-\:p\:-\:\frac{1}{q}}{p + 2}\)
\(\log_{18}{3.2} = \frac{4p\:-\:\frac{1}{q}}{p + 2}\)
\(\log_{18}{3.2} = \frac{4pq\:-\:1}{pq + 2q}\)
Salah
\(\log_{18}{3.2} = \frac{\log_{3}{3.2}}{\log_{3}{18}}\)
\(\log_{18}{3.2} = \frac{\log_{3}{\frac{32}{10}}}{\log_{3}{2\cdot 3^2}}\)
\(\log_{18}{3.2} = \frac{\log_{3}{32}\:-\:\log_{3}{10}}{\log_{3}{2}+\log_{3}{3^2}}\)
\(\log_{18}{3.2} = \frac{\log_{3}{2^5}\:-\:\log_{3}{2\cdot 5}}{\log_{3}{2}+\log_{3}{3^2}}\)
\(\log_{18}{3.2} = \frac{5\cdot\log_{3}{2}\:-\:\log_{3}{2}\:-\:\log_{3}{5}}{\log_{3}{2} + 2\cdot\log_{3}{3}}\)
\(\log_{18}{3.2} = \frac{5p\:-\:p\:-\:\frac{1}{q}}{p + 2}\)
\(\log_{18}{3.2} = \frac{4p\:-\:\frac{1}{q}}{p + 2}\)
\(\log_{18}{3.2} = \frac{4pq\:-\:1}{pq + 2q}\)
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Pertanyaan 4 dari 10
4. Pertanyaan
1 pointsIf \(a = 0.111111\dotso\), then \(\log_{a}{9\sqrt{3}} = \dotso\)
Benar
\(a = 0.111111\dotso\)
\(a = \frac{1}{9}\)
\(\log_{a}{9\sqrt{3}} = \log_{\frac{1}{9}}{3^{\frac{5}{2}}}\)
\(\log_{a}{9\sqrt{3}} = \log_{3^{-2}}{3^{\frac{5}{2}}}\)
\(\log_{a}{9\sqrt{3}} = \frac{\frac{5}{2}}{-2}\log_{3}{3}\)
\(\log_{a}{9\sqrt{3}} = -\frac{5}{4}\)
Salah
\(a = 0.111111\dotso\)
\(a = \frac{1}{9}\)
\(\log_{a}{9\sqrt{3}} = \log_{\frac{1}{9}}{3^{\frac{5}{2}}}\)
\(\log_{a}{9\sqrt{3}} = \log_{3^{-2}}{3^{\frac{5}{2}}}\)
\(\log_{a}{9\sqrt{3}} = \frac{\frac{5}{2}}{-2}\log_{3}{3}\)
\(\log_{a}{9\sqrt{3}} = -\frac{5}{4}\)
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Pertanyaan 5 dari 10
5. Pertanyaan
1 points\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} = \dotso\)
Benar
\(\text{Using } \color{red} \sqrt{(a + b) \pm 2\sqrt{ab}} = \sqrt{a} \pm \sqrt{b}\)
\(\sqrt{3\:-\:\sqrt{8}} = \sqrt{3\:-\:2\sqrt{2}} = \sqrt{2}\:-\:1\)
\(\sqrt{3 + \sqrt{8}} = \sqrt{3 + 2\sqrt{2}} = \sqrt{2} + 1\)
\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} = \log_{2^3}{(\sqrt{2}\:-\:1 + \sqrt{2} + 1)}\)
\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} = \log_{2^3}{2\sqrt{2}}\)
\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} = \log_{2^3}{2^{\frac{5}{2}}}\)
\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} = \frac{\frac{5}{2}}{3}\cdot \log_{2}{2}\)
\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} =\frac{5}{6}\)
Salah
\(\text{Using } \color{red} \sqrt{(a + b) \pm 2\sqrt{ab}} = \sqrt{a} \pm \sqrt{b}\)
\(\sqrt{3\:-\:\sqrt{8}} = \sqrt{3\:-\:2\sqrt{2}} = \sqrt{2}\:-\:1\)
\(\sqrt{3 + \sqrt{8}} = \sqrt{3 + 2\sqrt{2}} = \sqrt{2} + 1\)
\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} = \log_{2^3}{(\sqrt{2}\:-\:1 + \sqrt{2} + 1)}\)
\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} = \log_{2^3}{2\sqrt{2}}\)
\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} = \log_{2^3}{2^{\frac{5}{2}}}\)
\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} = \frac{\frac{5}{2}}{3}\cdot \log_{2}{2}\)
\(\log_{8}{(\sqrt{3\:-\:\sqrt{8}} + \sqrt{3 + \sqrt{8}})} =\frac{5}{6}\)
Hint
\(\text{Using } \color{red} \sqrt{(a + b) \pm 2\sqrt{ab}} = \sqrt{a} \pm \sqrt{b}\) -
Pertanyaan 6 dari 10
6. Pertanyaan
1 points\(\text{If } \log_{5p}{16} = \log_{4p}{25}, \text{ then } 20p = \dotso\)
Benar
\(\text{Let } \log_{5p}{16} = m \rightarrow 16 = (5p)^m\)
\(\text{Let } \log_{4p}{25} = m \rightarrow 25 = (4p)^m\)
\(16 = (5p)^m\)
\(16 = 5^m\cdot p^m\)
\(p^m = \frac{16}{5^m}\)
\(25 = (4p)^m\)
\(25 = 4^m\cdot p^m\)
\(25 = 4^m\cdot \frac{16}{5^m}\)
\(\frac{25}{16} = (\frac{4}{5})^m\)
\((\frac{5}{4})^2 = (\frac{4}{5})^m\)
\((\frac{4}{5})^{-2} = (\frac{4}{5})^m\)
\(m = -2\)
\(p^{-2} = \frac{16}{5^{-2}}\)
\(p^{-2} = \frac{16}{\frac{1}{25}}\)
\(p^{-2} = 400\)
\(\frac{1}{p^2} = 400\)
\(p^2 = \frac{1}{400}\)
\(p = \sqrt{\frac{1}{400}} = \frac{1}{20}\)
\(\text{So, } 20p = 20(\frac{1}{20}) = 1\)
Salah
\(\text{Let } \log_{5p}{16} = m \rightarrow 16 = (5p)^m\)
\(\text{Let } \log_{4p}{25} = m \rightarrow 25 = (4p)^m\)
\(16 = (5p)^m\)
\(16 = 5^m\cdot p^m\)
\(p^m = \frac{16}{5^m}\)
\(25 = (4p)^m\)
\(25 = 4^m\cdot p^m\)
\(25 = 4^m\cdot \frac{16}{5^m}\)
\(\frac{25}{16} = (\frac{4}{5})^m\)
\((\frac{5}{4})^2 = (\frac{4}{5})^m\)
\((\frac{4}{5})^{-2} = (\frac{4}{5})^m\)
\(m = -2\)
\(p^{-2} = \frac{16}{5^{-2}}\)
\(p^{-2} = \frac{16}{\frac{1}{25}}\)
\(p^{-2} = 400\)
\(\frac{1}{p^2} = 400\)
\(p^2 = \frac{1}{400}\)
\(p = \sqrt{\frac{1}{400}} = \frac{1}{20}\)
\(\text{So, } 20p = 20(\frac{1}{20}) = 1\)
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Pertanyaan 7 dari 10
7. Pertanyaan
1 pointsIf \(\log_{}{2} = 0.301\) and \(\log_{}{3} = 0.477\), then \(\log_{}{135} = \dotso\)
Benar
\(\log_{}{135} = \log_{}{(3^3\cdot 5)}\)
\(\log_{}{135} = \log_{}{3^3} + \log_{}{5}\)
\(\log_{}{135} = 3\cdot \log_{}{3} + \log_{}{5}\)
\(\log_{}{135} = 3\cdot \log_{}{3} + \log_{}{\frac{10}{2}}\)
\(\log_{}{135} = 3\cdot \log_{}{3} + \log_{}{10}\:-\:\log_{}{2}\)
\(\log_{}{135} = 3(0.477) + 1\:-\:(0.301)\)
\(\log_{}{135} = 2.13\)
Salah
\(\log_{}{135} = \log_{}{(3^3\cdot 5)}\)
\(\log_{}{135} = \log_{}{3^3} + \log_{}{5}\)
\(\log_{}{135} = 3\cdot \log_{}{3} + \log_{}{5}\)
\(\log_{}{135} = 3\cdot \log_{}{3} + \log_{}{\frac{10}{2}}\)
\(\log_{}{135} = 3\cdot \log_{}{3} + \log_{}{10}\:-\:\log_{}{2}\)
\(\log_{}{135} = 3(0.477) + 1\:-\:(0.301)\)
\(\log_{}{135} = 2.13\)
Hint
\(\color{red}\log_{}{5} = \log_{}{\frac{10}{2}}\) -
Pertanyaan 8 dari 10
8. Pertanyaan
1 points\(\log_{p}{\frac{1}{q^2}} \times \log_{q}{\frac{1}{r^3}} \times \log_{r}{\frac{1}{p^4}} = \dotso\)
Benar
\(\log_{p}{q^{-2}} \times \log_{q}{r^{-3}} \times \log_{r}{p^{-4}}\)
\(-2\cdot (-3) \cdot (-4) \cdot \log_{p}{q} \times \log_{q}{r} \times \log_{r}{p}\)
\(-24\cdot \log_{p}{p}\)
\(-24\cdot 1\)
\(-24\)
Salah
\(\log_{p}{q^{-2}} \times \log_{q}{r^{-3}} \times \log_{r}{p^{-4}}\)
\(-2\cdot (-3) \cdot (-4) \cdot \log_{p}{q} \times \log_{q}{r} \times \log_{r}{p}\)
\(-24\cdot \log_{p}{p}\)
\(-24\cdot 1\)
\(-24\)
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Pertanyaan 9 dari 10
9. Pertanyaan
1 points\(\text{If } 5^x = 3 \:-\: \sqrt{8}, \text{ then } \log_{3 + \sqrt{8}}{25^x} = \dotso\)
Benar
\(5^x = 3 \:-\: \sqrt{8}\)
\(5^x = \frac{1}{3 +\sqrt{8}}\rightarrow 3 + \sqrt{8} = 5^{-x}\)
\(\log_{3 + \sqrt{8}}{25^x} = \log_{5^{-x}}{5^{2x}}\)
\(\log_{3 + \sqrt{8}}{25^x} = \frac{2x}{-x}\cdot \log_{5}{5}\)
\(\log_{3 + \sqrt{8}}{25^x} = -2\cdot 1 = -2\)
Salah
\(5^x = 3 \:-\: \sqrt{8}\)
\(5^x = \frac{1}{3 +\sqrt{8}}\rightarrow 3 + \sqrt{8} = 5^{-x}\)
\(\log_{3 + \sqrt{8}}{25^x} = \log_{5^{-x}}{5^{2x}}\)
\(\log_{3 + \sqrt{8}}{25^x} = \frac{2x}{-x}\cdot \log_{5}{5}\)
\(\log_{3 + \sqrt{8}}{25^x} = -2\cdot 1 = -2\)
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Pertanyaan 10 dari 10
10. Pertanyaan
1 points\(\dfrac{\log_{3}{216}}{(\log_{3}{18})^2\:-\:(\log_{3}{2})^2} = \dotso\)
Benar
\(\dfrac{\log_{3}{216}}{(\log_{3}{18})^2\:-\:(\log_{3}{2})^2}\)
\(\dfrac{\log_{3}{216}}{(\log_{3}{18} + \log_{3}{2})(\log_{3}{18} \:-\: \log_{3}{2})}\)
\(\dfrac{\log_{3}{6^3}}{(\log_{3}{18\cdot 2})(\log_{3}{\frac{18}{2}})}\)
\(\dfrac{3\cdot \log_{3}{6}}{(\log_{3}{36})(\log_{3}{9})}\)
\(\dfrac{3\cdot \log_{3}{6}}{\log_{3}{6^2}\cdot \log_{3}{3^2}}\)
\(\dfrac{3\cdot \log_{3}{6}}{2\cdot \log_{3}{6}\cdot 2\cdot \log_{3}{3}}\)
\(\dfrac{3\cdot \cancel{\log_{3}{6}}}{2\cdot \cancel{\log_{3}{6}}\cdot 2\cdot 1}\)
\(\frac{3}{2\cdot 2}\)
\(\frac{3}{4}\)
Salah
\(\dfrac{\log_{3}{216}}{(\log_{3}{18})^2\:-\:(\log_{3}{2})^2}\)
\(\dfrac{\log_{3}{216}}{(\log_{3}{18} + \log_{3}{2})(\log_{3}{18} \:-\: \log_{3}{2})}\)
\(\dfrac{\log_{3}{6^3}}{(\log_{3}{18\cdot 2})(\log_{3}{\frac{18}{2}})}\)
\(\dfrac{3\cdot \log_{3}{6}}{(\log_{3}{36})(\log_{3}{9})}\)
\(\dfrac{3\cdot \log_{3}{6}}{\log_{3}{6^2}\cdot \log_{3}{3^2}}\)
\(\dfrac{3\cdot \log_{3}{6}}{2\cdot \log_{3}{6}\cdot 2\cdot \log_{3}{3}}\)
\(\dfrac{3\cdot \cancel{\log_{3}{6}}}{2\cdot \cancel{\log_{3}{6}}\cdot 2\cdot 1}\)
\(\frac{3}{2\cdot 2}\)
\(\frac{3}{4}\)
Hint
\(\color{red} x^2 \:-\: y^2 = (x + y)(x\:-\:y)\)