Haii adik-adik...
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!
41. \(\textrm{Diketahui }^{5}\log 3=a \textrm{ dan }^{3}\log 4=b.\textrm{ Nilai }^{4}\log 15=...\)
a) \(\frac{1+a}{ab}\)
b) \(\frac{1+a}{a+b}\)
c) \(\frac{1+a}{a-b}\)
d) \(\frac{ab}{1-a}\)
e) \(\frac{ab}{1-b}\)
\(^{5}\log 3=a;^{3}\log 4=b\\\textrm{Maka:}^{4}\log 15=\frac{\log 15}{\log 4}=\frac{\log 5.3}{\log 4}=\frac{\log 5+\log 3}{\log 4}=\frac{^{3}\log 5+^{3}\log 3}{^{3}\log 4}=\frac{\frac{1}{a}+1}{b}=\frac{1+a}{a}\times \frac{1}{b}=\frac{1+a}{ab}\)
42. \(\textrm{Jika persamaan }^{x}\log \left ( 2 \right )+^{x}\log \left ( 3x-4 \right )=2\\ \textrm{ mempunyai akar-akar }x_{1}\textrm{ dan }x_{2}\textrm{ dengan }x_{1}> x_{2}\textrm{ maka }x_{1}-x_{2}=...\)
a) \(0\)
b) \(1\)
c) \(2\)
d) \(3\)
e) \(4\)
\(^{x}\log \left ( 2 \right )+^{x}\log \left ( 3x-4 \right )=2\\\Rightarrow ^{x}\log\left ( 2 \right )\left ( 3x-4 \right )=^{x}\log x^{2}\\\Rightarrow \left ( 2 \right )\left ( 3x-4 \right )=x^{2}\\\Rightarrow 6x-8+x^{2}\\\Rightarrow x^{2}-6x+8=0\\\Rightarrow \left ( x-4 \right )\left ( x-2 \right )=0\\\Rightarrow x_{1}=4\textrm{ dan }x_{2}=2\left ( x_{1}> x_{2} \right )\\\textrm{Jadi, nilai dari }x_{1}-x_{2}=4-2=2\)
43. \(\textrm{Persamaan }^{\left ( x^{2}-6x+14 \right )}\log \left ( x-3 \right )=^{\left ( 4x^{2}-4x+1 \right )}\log \left ( x^{2}-6x+9 \right )\textrm{ dipengaruhi oleh x =...}\)
a) \(6\)
b) \(3\textrm{ atau }5\)
c) \(3\)
d) \(5\)
e) \(6\)
\(^{\left ( x^{2}-6x+14 \right )}\log\left ( x-3 \right )=^{\left ( 4x^{2}-4x+1 \right )}\log\left ( x^{2}-6x+9 \right )\\\Rightarrow ^{\left ( x^{2}-6x+14 \right )}\log\left (x- 3\right )=^{\left ( 2x-1 \right )^{2}}\log\left ( x-3 \right )^{2}\\\Rightarrow ^{\left ( x^{2-6x+14} \right )}\log\left ( x-3 \right )=^{\left ( 2x-1 \right )}\log\left ( x-3 \right )\\\Rightarrow x^{2}-6x+14=2x-1\\\Rightarrow x^{2}-8x+15=0\\\Rightarrow \left ( x-3 \right )\left ( x-5 \right )=0\\x=3\textrm{ atau }x=5\\\textrm{Syarat domain }x-3> 0\textrm{ maka HP={5}}\)
44. \(\textrm{Nilai x yang memenuhi }\frac{2x^{\log4x}}{x^{\log2x}}< \frac{1}{2}\textrm{ adalah ....}\)
a) \(x< -100\)
b) \(x< -10\)
c) \(0< x< \frac{1}{100}\)
d) \(\frac{1}{100}< x< \frac{1}{10}\)
e) \(2< x< 10\)
\(\frac{2x^{\log4x}}{x^{\log2x}}< \frac{1}{2}\\x^{\log4x-\log2x}< \frac{1}{4}\\x^{\log\frac{4x}{2x}}< \frac{1}{4}\\x^{\log2}< 2^{-2}\\ \log x^{\log2}< \log2^{-2}\\ \log2.\log x< -2.\log2\\ \log x< -2\Rightarrow \log x< \log 10^{-2}\\\Rightarrow x< \frac{1}{100}\\\textrm{Syarat domain untuk logaritma, yaitu }x> 0\\\textrm{Jadi, HP}=\left \{ 0< x< \frac{1}{100} \right \}\)
45. \(\textrm{Jika }b=a^{4}, a\textrm{ dan }b\textrm{ positif maka }^{a}\log b-^{b}\log a\textrm{ adalah...}\)
a) \(0\)
b) \(1\)
c) \(2\)
d) \(3\frac{3}{4}\)
e) \(4\frac{1}{4}\)
\(\textrm{Jika }b=a^{4}\textrm{ maka }^{a}\log b=4\textrm{ dan }^{b}\log a=\frac{1}{4}\\\textrm{Jadi, nilai dari }^{a}\log b-^{b}\log a=4-\frac{1}{4}=3\frac{3}{4}\)
46. \(\textrm{Jika }a=^{9}\log\left ( \sqrt[3]{16} \right )\textrm{ dan }b=^{2}\log\left ( \frac{1}{3} \right )\textrm{ maka ab =...}\)
a) \(\frac{4}{3}\)
b) \(\frac{2}{3}\)
c) \(\frac{4}{9}\)
d) \(-\frac{2}{3}\)
e) \(-\frac{4}{3}\)
\(a=^{9}\log\left ( \sqrt[3]{16} \right )\textrm{ dan }b=^{2}\log\left ( \frac{1}{3} \right )\\\textrm{Maka:}\\ab=^{9}\log\left ( \sqrt[3]{16} \right ).^{2}\log\left ( \frac{1}{3} \right )=^{3^{2}}\log^{\frac{4}{3}}.^{2}\log3^{-1}=\frac{4}{\frac{3}{2}}^{3}\log 2.^{2}\log 3^{-1}=\frac{2}{3}^{3}\log 3^{-1}=\frac{2}{3}.\left ( -1 \right )=-\frac{2}{3}\)
47. \(\textrm{Nilai maksimum fungsi }f\left ( x \right )=^{2}\log\left ( x+5 \right )+^{2}\log\left ( 3-x \right )\textrm{ adalah ...}\)
a) \(4\)
b) \(8\)
c) \(12\)
d) \(15\)
e) \(16\)
\(f\left ( x \right )=^{2}\log\left ( x+5 \right )+^{2}\log\left ( 3-x \right )\\\Rightarrow f\left ( x \right )=^{2}\log\left ( x+5 \right )\left ( 3-x \right )\\\Rightarrow f\left ( x \right )=^{2}\log\left ( -x^{2}-2x+15 \right )\\\textrm{Misalkan, }-x^{2}-2x+15=g\left ( x \right )\textrm{ maka }f\left ( x \right )\textrm{ akan maksimal jika }g\left ( x \right )\textrm{ juga maksimal.}\\\textrm{Nilai maksimal dari g(x) jika: }\\x=-\frac{b}{2a}=-\frac{-2}{2\left ( -1 \right )}=-1\\\textrm{Sehingga nilai maksimal dari f(x) adalah: }\\f\left ( -1 \right )=^{2}\log\left ( -\left ( -1 \right )^{2}-2\left ( -1 \right )+15 \right )=^{2}\log\left ( -1+2+15 \right )=^{2}\log 16=4\)
48. \(\textrm{Nilai dari }\frac{\log\left ( a^{2}-x^{2} \right )}{\log a}-^{a}\log\left [ 1-\frac{x^{2}}{a^{2}} \right ]\textrm{ adalah...}\)
a) \(-2\)
b) \(-1\)
c) \(1\)
d) \(2\)
e) \(3\)
\(\frac{\log\left ( a^{2}-x^{2} \right )}{\log a}-^{a}\log\left [ 1-\frac{x^{2}}{a^{2}} \right ]=^{a}\log\left ( a^{2}-x^{2} \right )-^{a}\log\left [ 1-\frac{x^{2}}{a^{2}}\right ]=^{a}\log \left ( \frac{a^{2}-x^{2}}{1-\frac{x^{2}}{a^{2}}} \right )\\ =^{a}\log\left ( \frac{a^{2}-x^{2}}{\frac{a^{2}}{a^{2}}-\frac{x^{2}}{a^{2}}} \right )=^{a}\log\left ( a^{2}-x^{2}:\frac{a^{2}-x^{2}}{a^{2}} \right )=^{a}\log\left ( a^{2}-x^{2}\times \frac{a^{2}}{a^{2}-x^{2}} \right )=^{a}\log\left ( a^{2} \right )=2\)
49. \(\textrm{Jika }^{\frac{1}{a}}\log\frac{1}{b}=2\textrm{ maka...}\)
a) \(^{b}\log a=2\)
b) \(^{a}\log b=2\)
c) \(^{\frac{1}{a}}\log\frac{1}{b}=\frac{1}{2}\)
d) \(^{a}\log\frac{1}{b}=2\)
e) \(^{b}\log\frac{1}{a}=\frac{1}{2}\)
\(^{\frac{1}{a}}\log\frac{1}{b}=2\\\Rightarrow ^{a^{-1}}\log b^{-1}=2\\\Rightarrow \frac{-1}{-1}^{a}\log b=2\Rightarrow ^{a}\log b=2\)
50. \(\textrm{Jika }\left ( ^{a}\log\left ( 3x-1 \right ) \right )-\left ( ^{5}\log a \right )=3\textrm{ maka nilai x adalah...}\)
a) \(36\)
b) \(39\)
c) \(42\)
d) \(45\)
e) \(48\)
\(\left ( ^{a}\log\left ( 3x-1 \right ) \right )\left ( ^{5}\log a \right )=3\\\Rightarrow ^{a}\log\left ( 3x-1 \right )=\frac{3}{^{5}\log a}\\\Rightarrow ^{a}\log\left ( 3x-1 \right )=3.^{a}\log 5\\\Rightarrow ^{a}\log\left ( 3x-1 \right )=^{a}\log5^{3}\\\Rightarrow 3x-1=125\\\Rightarrow x=\frac{125+1}{3}=42\)
