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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!
11. \(\textrm{Nilai x yang memenuhi persamaan }10^{4\log x}-5(10)^{2\log x}=-4\textrm{ adalah...}\)
a) \(1\)
b) \(4\)
c) \(1\textrm{ atau }2\)
d) \(1\textrm{ atau }4\)
e) \(2\textrm{ atau }4\)
\(\left [ 10^{2\log x} \right ]^{2}-5\left [ 10^{2\log x} \right ]+4=0\\\left [ 10^{2\log x} -4\right ]\left [ 10^{2\log x} -4\right ]=0\textrm{ syarat:}\log x\textrm{ ada jika }x > 0\\2\log x=\log 4\rightarrow x^{2}=4\textrm{ maka }x=2\\\square \left [ 10^{2\log x}-1 \right ]=0\\2\log x=\log 1\rightarrow x^{2}=1\textrm{ maka }x=1\)
12. \(\textrm{Himpunan penyelesaian dari }\log(x^{2}+4x+4)\leq \log(5x+10)\textrm{ adalah....}\)
a) \(-2<x\leq 3\)
b) \(x<3\)
c) \(-3<x< 2\)
d) \(x\leq -2\textrm{ atau }x\geq 3\)
e) \(2\leq x\leq 3\)
\(\log(x^{2}+4x+4)\leq \log(5x+10)\rightarrow (x^{2}+4x+4)\leq (5x+10)\Rightarrow (x-3)(x+2)\leq 0\\\textrm{Tanda }\leq \textrm{ pasti penyelesaian di antara pembuat nol :}-2<x<3,x\neq -2\)
13. \(\textrm{Nilai-nilai x yang memenuhi }0\leq x\leq \pi \textrm{ dan }^{2}\log^{2}(\sin x)-2\log(\sin ^{3}x)\leq 4\textrm{ adalah....}\)
a) \(0\leq x\leq \frac{\pi }{6}\)
b) \(\frac{\pi }{6}\leq x\leq \pi \)
c) \(\frac{\pi }{6}\leq x\leq \frac{5\pi }{6}\)
d) \(\frac{5\pi }{6}\leq x\leq \pi \)
e) \(\frac{\pi }{6}\leq x\leq \frac{\pi }{3}\)
\(^{2}\log^{2}(\sin x)-^{2}\log (\sin ^{3}x)\leq 4\\\Rightarrow ^{2}\log^{2}(\sin x)-3\cdot ^{2}\log(\sin x)-4\leq 0\\\textrm{Misalka, }p=^{2}\log(\sin x)\\\Rightarrow p^{2}-3p-4\leq 0\\\Rightarrow (p-4)(p+1)\leq 0\\\Rightarrow -1\leq p\leq \leq 4\\\Rightarrow -1\leq ^{2}\log(\sin x)\leq 4\\\Rightarrow ^{2}\log2^{-1}\leq ^{2}\log(\sin x)\leq ^{2}\log2^{4}\\\Rightarrow \frac{1}{2}\leq \sin x\leq 16\\\Rightarrow \frac{\pi }{6}\leq x\leq \frac{5\pi }{6}\)
14. \(\textrm{Jika}\ X_{1}\ \textrm{dan}\ X_{2}\ \textrm{adalah akar-akar persamaan }\log \left ( x^{2}+7x+20 \right )=1\\ \textrm{maka }\left ( x_{1}+x_{2} \right )^{2}-4x_{1}x^{2}\ \textrm{adalah...}\)
a) \(49\)
b) \(29\)
c) \(20\)
d) \(19\)
e) \(9\)
\(\log\left ( x^{2}+7x+20 \right )=1\times \log 10\\\rightarrow x^{2}+7x+20=10\\\rightarrow x^{2}+7x+20=0\\\rightarrow \left ( x+5 \right )\left ( x+2 \right )=0\\\rightarrow x_{1}=-5\ x_{2}=-2\\\Rightarrow \left ( x_{1}+x_{2} \right )^{2}-4x_{1}x_{2}=\left ( -5+-2 \right )^{2}-4\left ( -5\times -2 \right )=\left ( -7 \right )^{2}-4\times 10=49-40=9\)
15. \(^{20}\log 3=p\textrm{ dan }^{50}\log 3=q\textrm{ maka }^{3}\log 100=....\)
a) \(\frac{2p+2q}{3pq}\)
b) \(\frac{p^{3}+q^{3}}{pq}\)
c) \(\left ( \frac{1}{p} +\frac{1}{q}\right )^{3}\)
d) \(\frac{3}{p} +\frac{3}{q}\)
e) \(\frac{p+q}{pq}\)
\(\textrm{Diketahui :}\\^{20}\log3=p\textrm{ dan }^{5}\log3=q\\\textrm{ Maka:}^{3}\log20=\frac{1}{p}\textrm{ dan }^{3}\log5=\frac{1}{q}\\\textrm{Sehingga:}\\^{3}\log100=\frac{\log100}{\log3}=\frac{\log20.5}{\log3}=\frac{\log20+\log5}{\log3}=\frac{^{3}\log20+^{3}\log5}{^{3}\log3}=\frac{\frac{1}{p}+\frac{1}{q}}{1}=\frac{p+q}{pq}\)
16. \(\textrm{Himpunan penyelesaian pertidaksamaan }2^{4x}-2^{2(x+1)}+3<0\textrm{ adalah...}\)
a) \(\left \{ x\mid 1<x<3 \right \}\)
b) \(\left \{ x\mid 0<x<^{3}\log\sqrt{2} \right \}\)
c) \(\left \{ x\mid x<0\textrm{ atau } x<^{2}\log\sqrt{3}\right \}\)
d) \(\left \{ x\mid 0<x<^{2}\log\sqrt{3} \right \}\)
e) \(\left \{ x\mid 0<x<^{2} \log3\right \}\)
\(\textrm{Diketahui }\\2^{4x}-2^{2(x+1)}+3<0\\\Rightarrow 2^{4x}-2^{2x+2}+3<0\\\Rightarrow 2^{4x}-2^{2}\cdot 2^{2x}+3<0\\\textrm{Misal :}2^{2x}=p\\\Rightarrow p^{2}-4p+3<0\\\Rightarrow (p-3)(p-1)<0\\\textrm{Diperoleh batas p=3 atau p=1}\\\bullet p=3\Rightarrow 2^{2x}=3\\\Rightarrow 2^{x}=3^{\frac{1}{2}}\Rightarrow 2^{x}=\sqrt{3}\Rightarrow x=^{2}\log\sqrt{3}\\\bullet p=1\Rightarrow 2^{2x}=1\\\Rightarrow 2^{2x}=2^{0}\Rightarrow x=0\\Hp=\left \{ x\mid x<0\textrm{ atau }x<^{2}\log\sqrt{3} \right \}\)
17. \(\textrm{Persamaan }^{x}\log 2+^{x}\log(3x-4)=2\textrm{ mempunyai dua penyelesaian, yaitu }x_{1}\textrm{ dan }x_{2}.\textrm{Nilai dari }x_{1}\cdot x_{2}\textrm{ adalah....}\)
a) \(8\)
b) \(6\)
c) \(4\)
d) \(3\)
e) \(2\)
\(\textrm{Diketahui :}\\^{x}\log 2+^{x}\log(3x-4)=^{x}\log x^{2}\\\Rightarrow ^{x}\log2(3x-4)=^{x}\log x^{2}\\\Rightarrow 6x-8=x^{2}\\\Rightarrow x^{2}-6x+8=0\\\textrm{Mempunyai akar- akar }x_{1}\textrm{ dan }x_{2}\textrm{ maka:}x_{1}\cdot x_{2}=\frac{c}{a}=8\)
18. \(\textrm{Nilai }\left ( ^{a}\log\frac{1}{b^{2}} \right )\left ( ^{b}\log\frac{1}{c^{2}} \right )\left ( ^{c}\log\frac{1}{a^{2}} \right )=....\)
a) \(-14\)
b) \(-12\)
c) \(-10\)
d) \(-8\)
e) \(-6\)
\(\left ( ^{a}\log\frac{1}{b^{2}} \right )\left ( ^{b}\log\frac{1}{c^{2}} \right )\left ( ^{c}\log\frac{1}{a^{2}} \right )\\=\left ( ^{a}\log b^{-2} \right )\left ( ^{b}\log c^{-2} \right )\left ( ^{c}\log a^{-2} \right )\\=(-2)\cdot (-2)\cdot (-2)\cdot ^{a}\log b\cdot ^{b}\log c\cdot ^{c}\log a\\=-8\cdot ^{a}\log a=-8\)
19. \(\textrm{Jika }X_{1}\textrm{ dan }X_{2}\textrm{ memenuhi persamaan }\frac{^{10}\log \frac{x^{5}}{10}}{^{10}\log x}-^{10}\log x=\frac{5}{^{10}\log x},\textrm{ maka }X_{1}+X_{2}=...\)
a) \(5\)
b) \(6\)
c) \(60\)
d) \(110\)
e) \(1100\)
\(\underline{\frac{^{10}\log \frac{x^{5}}{10}}{^{10}\log x}-^{10}\log x= \frac{5}{^{10}\log x}}\Rightarrow \times ^{10}\log x\\\log \frac{x^{5}}{10}-\left ( \log x \right )^{2}=5\\\log x^{5}-\log 10-\left ( \log x \right )^{2}=5\\5\log x-1-\left ( \log x \right )^{2}-5=0\\5\log x-6-\left ( \log x \right )^{2}=0\\\left ( \log x \right )^{2}-5\log x+6\\\left ( \log x-3 \right )\left ( \log x-2 \right )=0\\^{10}\log x=3=X_{1}=1000\textrm{ atau }^{10}\log x=2=X_{2}=100\\\textrm{Jadi,}X_{1}+X_{2}=1000+100=1100\)
20. \(\textrm{Diketahui }2(^{4}\log x)^{2}-2^{4}\log \sqrt{x}=1.\\\textrm{Jika akar-akar persamaan di atas adalah }x_{1}\textrm{ dan }x_{2}\textrm{ maka }x_{1}+x_{2}=....\)
a) \(5\)
b) \(4\frac{1}{2}\)
c) \(4\frac{1}{4}\)
d) \(2\frac{1}{2}\)
e) \(2\frac{1}{4}\)
\(2(^{4}\log x)^{2}-2^{4}\log \sqrt{x}=1\\\Rightarrow 2(^{4}\log x)^{2}-2^{4}\log x^{\frac{1}{2}}-1=0\\\Rightarrow 2(^{4}\log x)^{2}-^{4}\log x-1=0\\\textrm{Misalkan :}^{4}\log x=p\textrm{ sehingga menjadi :}2p^{2}-p-1=0\\\Rightarrow (2p+1)(p-1)=0\\\Rightarrow p_{1}=-\frac{1}{2}\textrm{ dan }p_{2}=1\\\bullet p_{1}=-\frac{1}{2}\Rightarrow ^{4}\log x=-\frac{1}{2}\\\Rightarrow x_{1}=4^{-\frac{1}{2}}\Rightarrow \frac{1}{\sqrt{4}}=\frac{1}{2}\\\bullet p_{2}=1\Rightarrow ^{4}\log x=1\\\Rightarrow x_{2}=4\\\textrm{Jadi,}x_{1}+x_{2}=4\frac{1}{2}\)
