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Selamat datang di halaman contoh soal dan pembahasan "Logaritma". Di halaman ini akan membahas tentang contoh soal dan pembahasan lengkap mengenai Logaritma.
Untuk lebih jelasnya mari kita lihat contoh soal dan pembahasan dibawah ini!
31. \(\textrm{Nilai dari }^{9}\log 25\times ^{5}\log 2-^{3}\log 54=...\)
a) \(-3\)
b) \(-1\)
c) \(0\)
d) \(2\)
e) \(3\)
\(^{9}\log 25\times ^{5}\log 2-^{3}\log 54\\=^{3^{2}}\log 5^{2}\times ^{5}\log 2-^{3}\log 54\\=\frac{2}{2}\times ^{3}\log 5\times ^{5}\log 2-^{3}\log 54\\=^{3}\log 2-^{3}\log 54\\=^{3}\log \frac{2}{54}\\=^{3}\log \frac{1}{27}=^{3}\log \frac{1}{3^{3}}=^{3}\log 3^{-3}=-3\)
32. \(\textrm{Jika }^{25}\log 5^{2x}=8\textrm{ maka x=...}\)
a) \(\frac{1}{4}\)
b) \(\frac{1}{2}\)
c) \(6\)
d) \(8\)
e) \(10\)
\(\textrm{Diketahui: }^{25}\log 5^{2x}=8\textrm{ maka: }^{5^{2}}\log 5^{2x}=8\\\Rightarrow \frac{2x}{2}\times ^{5}\log 5=8\\\Rightarrow x\times 1=8\Rightarrow x=8\)
33. \(\textrm{Nilai x yang memenuhi persamaan: }^{\frac{1}{2}}\log \left ( x^{2} -3\right )-^{\frac{1}{2}}\log x =-1\textrm{ adalah ...}\)
a) \(\textrm{x=-1 atau x=3}\)
b) \(\textrm{x=1 atau x=-3}\)
c) \(\textrm{x=1 atau x=3}\)
d) \(\textrm{x=1 saja}\)
e) \(\textrm{x= 3saja}\)
\(^{\frac{1}{2}}\log \left ( x^{2}-3 \right )-^{\frac{1}{2}}\log x=-1\\\Rightarrow ^{\frac{1}{2}}\log \left ( x^{2}-3 \right )-^{\frac{1}{2}}\log x =^{\frac{1}{2}}\log \left ( \frac{1}{2} \right )^{-1}\\\Rightarrow \frac{\left ( x^{2}-3 \right )}{x}=\left ( \frac{1}{2} \right )^{-1}\\\Rightarrow \frac{x^{2}-3}{x}=2\\\Rightarrow x^{2}-3=2x\\\Rightarrow x^{2}-2x-3=0\\\Rightarrow \left ( x+1 \right )\left ( x-3 \right )=0\\\Rightarrow x=-1\textrm{ atau }x=3\\x=-1\left ( \textrm{tidak mungkin } \right )\\\textrm{Jadi, x=3 saja.}\)
34. \(\textrm{Nilai dari: }^{\frac{1}{2}}\log 5\times ^{5}\log 4\times ^{2}\log \frac{1}{8}\times \left ( ^{5}\log 25 \right )^{2}=...\)
a) \(24\)
b) \(12\)
c) \(8\)
d) \(-4\)
e) \(-121\)
\(^{\frac{1}{2}}\log 5\times ^{5}\log 4\times ^{2}\log \frac{1}{8}\times \left ( ^{5} \log 25\right )^{2}\\=^{2^{-1}}\log 5\times ^{5}\log 2^{2}\times ^{2}\log 2^{-3}\times \left ( ^{5}\log 5^{2} \right )^{2}\\=-1\times ^{2}\log 2^{2}\times \left ( -3 \right )\times \left ( 2 \right )^{2}\\=-1\times 2\times \left ( -3 \right )\times 4=24\)
35. \(\textrm{Hasil dari }^{\frac{1}{5}}\log 625+^{64}\log \frac{1}{16}+4^{3^{25}\log 5}=...\)
a) \(-4\frac{19}{24}\)
b) \(3\frac{1}{3}\)
c) \(4\frac{2}{3}\)
d) \(5\frac{1}{3}\)
e) \(59\frac{1}{3}\)
\(\textrm{Beberapa sifat logaritma: }\\^{a^{m}}\log b^{n}=\frac{n}{m}^{a}\log b\\^{\frac{1}{5}}\log 625+^{64}\log \frac{1}{16}+4^{3.^{25}\log 5}\\=^{5^{-1}}\log 5^{4}+^{2^{6}}\log 2^{-4}+2^{2.3^{5^{2}\log 5}}\\=\frac{4}{-1}^{5}\log 5+\left ( \frac{-4}{6} \right )^{2}\log 2+2^{6.\frac{1}{2}^{5}\log 5}\\=-4.1-\frac{2}{3}+2^{3}=-4-\frac{2}{3}+8=3\frac{1}{3}\)
36. \(\textrm{Diketahui }^{64}\log\sqrt{16^{x-4}}=\frac{1}{2}.\textrm{ Nilai x yang memenuhi persamaan itu adalah...}\)
a) \(-5\frac{1}{2}\)
b) \(-4\frac{1}{3}\)
c) \(4\)
d) \(5\frac{1}{2}\)
e) \(9\frac{1}{2}\)
\(\textrm{Nilai x dari persamaan }^{64}\log \sqrt{16^{x-4}}=\frac{1}{2}\\\Rightarrow ^{64}\log \sqrt{16^{x-4}}=^{64}\log 64^{\frac{1}{2}}\\\Rightarrow \sqrt{16^{x-4}}=\sqrt{64}\\\Rightarrow 16^{x-4}=64\\\Rightarrow 2^{4(x-4)}=2^{6}\\\Rightarrow 4x-16=6\\\Rightarrow 4x=22\\\Rightarrow x=\frac{22}{4}=5\frac{1}{2}\)
37. \(\textrm{Nilai dari }\frac{^{3}\log 64.^{\sqrt{2}}\log 36}{^{9}\log 6}=...\)
a) \(6\)
b) \(12\)
c) \(16\)
d) \(24\)
e) \(48\)
\(\frac{^{3}\log 64.^{\sqrt{2}}\log 36}{^{9}\log 6}=\frac{^{3}\log 2^{6}.^{2^{\frac{1}{2}\log 6^{2}}}}{^{3^{2}\log 6}}=\frac{6.^{3}\log 2.4.^{2}\log 6}{\frac{1}{2}^{3}\log 6}=\frac{24.^{3}\log 6}{\frac{1}{2}^{3}\log 6}=\frac{24}{\frac{1}{2}}=48\)
38. \(\textrm{Akar-akar persamaan }^{4}\log \left ( 2x^{2}-3x+7 \right )=2\textrm{ adalah }x_{1}\textrm{ dan }x_{2}.\textrm{Nilai dari }4.x_{1}.x_{2}=...\)
a) \(-6\)
b) \(-18\)
c) \(10\)
d) \(18\)
e) \(46\)
\(\textrm{Diketahui:}\\\textrm{Akar-akar persamaan}\\^{4}\log \left ( 2x^{2}-3x+7 \right )=2\textrm{adalah }x_{1}\textrm{ dan }x_{2}\\\Rightarrow ^{4}\log \left ( 2x^{2}-3x+7 \right )=^{4}\log 4^{2}\\\Rightarrow \left ( 2x^{2}-3x+7 \right )=4^{2}\\\Rightarrow 2x^{2}-3x+7-16=0\\\Rightarrow 2x^{2}-3x-9=0\\\Rightarrow \left ( 2x+3 \right )\left ( x-3 \right )=0\\\Rightarrow x_{1}=-\frac{3}{2}\textrm{ dan }x_{2}=3\\\textrm{Jadi, nilai dari }4.x_{1}.x_{2}=4\left ( -\frac{3}{2} \right ).3=-18\)
39. \(\textrm{Diketahui: }^{2}\log 3=x\textrm{ dan }^{2}\log 10 =\textrm{y}.\textrm{ Nilai }^{6}\log 120=...\)
a) \(\frac{x+y+2}{x+1}\)
b) \(\frac{x+1}{x+y+2}\)
c) \(\frac{x}{xy+2}\)
d) \(\frac{xy+2}{x}\)
e) \(\frac{2xy}{x+1}\)
\(\textrm{Diketahui:}\\^{2}\log 3=x;^{2}\log 10=y\\\textrm{Maka, }^{6}\log 120=\frac{\log 120}{log 6}=\frac{\log 2^{2}.3.10}{\log 2.3}=\frac{\log 2^{2}+\log 3+\log 10}{\log 2+\log 3}=\frac{^{2}\log 2^{2}+^{2}\log 3}+^{2}\log 10{^{2}\log 2+^{2\log 3}}=\frac{2+x+y}{1+x}=\frac{x+y+2}{x+1}\)
40. \(\textrm{Diketahui }\log 2=0,3010\textrm{ dan }\log 3=0,4771\textrm{ maka }\log \left ( \sqrt[3]{2}\times \sqrt{3} \right )=...\)
a) \(0,1505\)
b) \(1590\)
c) \(2007\)
d) \(0,3389\)
e) \(3891\)
\(\log 2=0,3010\textrm{ dan }\log 3=0,4771\\\textrm{maka }\log \left ( \sqrt[3]{2} \times \sqrt{3}\right )=\log 2^{\frac{1}{3}}.3^{\frac{1}{2}}\\=\log 2^{\frac{1}{3}}+\log 3^{\frac{1}{2}}\\=\frac{1}{3}.\log 2+\frac{1}{2}.\log 3\\=\frac{1}{3}.\left ( 0,3010 \right )+\frac{1}{2}.\left ( 0,4771 \right )\\=0,10033+0,23855\\=0,33888\approx 0,3389\)
