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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!
21. \(\frac{\left ( ^{5}\log 10 \right )^{2}-\left ( ^{5}\log 2 \right )^{2}}{^{5}\log \sqrt{20}}=....\)
a) \(\frac{1}{2}\)
b) \(1\)
c) \(2\)
d) \(4\)
e) \(5\)
\(\frac{\left ( ^{5}\log 10 \right )^{2}-\left ( ^{5}\log 2 \right )^{2}}{^{5}\log\sqrt{20}}\\=\frac{\left ( ^{5}\log10+^{5}\log2 \right )\left ( ^{5}\log10-^{5}\log2 \right )}{^{5}\log\sqrt{20}}\\=\frac{\left ( ^{5}\log20 \right )\left ( ^{5}\log5 \right )}{^{5}\log\sqrt{20}}\\=\frac{\left ( ^{5}\log20 \right )\cdot 1}{^{5}\log20^{\frac{1}{2}}}\\=\frac{^{5}\log20}{\frac{1}{2}\cdot ^{5}\log20}=2\)
22. \(\textrm{Nilai maksimum fungsi:}\\f(x)=^{2}\log(x+5)+^{2}\log(3-x)\textrm{ adalah....}\)
a) \(4\)
b) \(8\)
c) \(12\)
d) \(15\)
e) \(16\)
\(\textrm{Diketahui:}\\f(x)=^{2}\log(x+5)+^{2}\log(3-x)\\=^{2}\log(x+5)(3-x)\\=^{2}\log(-x^{2}+2x+15)\\\textrm{Misalka :}-x^{2}+2x+15=\textrm{G(x)}\\ \textrm{ maka f(x)akan mencapai maksimum}\\ \textrm{ jika G(x) mencapai nilai maksimum.}\\\textrm{G(x) maksimum ketika x=}\frac{-b}{2a}=1,\textrm{yaitu:}\\G(x)=-(1)^{2}+2(1)+15=16\\\textrm{Sehingga f(x) maksimum ketika }\\f(x)=^{2}\log16=4\)
23. \(\textrm{Jika }\frac{^{2}\log a}{^{3}\log b}=m\textrm{ dan }\frac{^{3}\log a}{^{2}\log b}=n, a> 1\textrm{ dan }b> 1\textrm{ maka }\frac{m}{n}=\)
a) \(^{2}\log 3\)
b) \(^{3}\log 2\)
c) \(^{4}\log 9\)
d) \(\left ( ^{3}\log 2 \right )^{2}\)
e) \(\left ( ^{2}\log 3 \right )^{2}\)
\(\frac{m}{n}=\left ( \frac{^{2}\log a}{^{3}\log b} \right )\times \left ( \frac{^{2}\log b}{^{3}\log a} \right )=\frac{^{2}\log a}{^{3}\log b}\frac{^{a}\log 3}{^{b}\log 2}=\frac{^{2}\log 3}{^{3}\log 2}=\left ( ^{2}\log 3 \right )^{2}\)
24. \( \textrm{Nilai yang memenuhi persamaan adalah }^{2}\log \left ( 2x-3 \right )-^{4}\log \left ( x-\frac{3}{2} \right )=1\textrm{ adalah ...} \)
a) \(\frac{3}{2}\)
b) \(\frac{2}{3}\)
c) \(\frac{5}{2}\)
d) \(\frac{2}{5}\)
e) \(-\frac{2}{5}\)
\(^{2}\log \left ( 2x-3 \right )-^{4}\log\left ( x-\frac{3}{2} \right )=1\\^{2}\log\left ( \frac{2x-3}{\left ( x-\frac{3}{2} \right )^{\frac{1}{2}}} \right )=^{2}\log 2\\\left ( 2x-3 \right )=2\left ( x-\frac{3}{2} \right )^{\frac{1}{2}}\textrm{(dikuadratkan)}\\4x^{2}-12x+9=4x-6\\4x^{2}-16x+15=0\rightarrow \left ( 2x-3 \right )\left ( 2x-5 \right )=0\\x=\frac{3}{2}\textrm{ atau }x=\frac{5}{2}\rightarrow \textrm{ syarat }x-\frac{3}{2}> 0:x> \frac{3}{2}\textrm{ berarti yang memenuhi }x=\frac{5}{2}\)
25. \(\log x=\frac{1}{3}\log 8+\log 9-\frac{1}{3}\log 27\textrm{ dipengaruhi untuk x sama dengan...}\)
a) \(8\)
b) \(6\)
c) \(4\)
d) \(2\)
e) \(1\)
\(\log x=\log \left ( 8 \right )^{\frac{1}{3}}+\log 9-\log \left ( 27 \right )^{\frac{1}{3}}=\log \left ( \frac{2\times 9}{3} \right )=\log 6\rightarrow x=6\)
26. \(^{a}\log \frac{1}{b}\times ^{b}\log \frac{1}{c^{2}}\times ^{c}\log \frac{1}{a^{2}}=...\)
a) \(-6\)
b) \(6\)
c) \(\frac{b}{a^{2}c}\)
d) \(\frac{a^{2c}}{b}\)
e) \(-\frac{1}{6}\)
\(^{a}\log b^{-1}\times ^{b}\log c^{-2}\times ^{c}\log a^{-3}=\left ( -^{a} \log b\right )\left ( -2^{b}\log c \right )\left ( -3^{c} \log a\right )=-6 ^{a}\log a=-6\)
27. \(\textrm{Jumlah akar-akar persamaan }\log \left (\frac{x^{2}+16}{x} \right )=1 \textrm{ adalah...}\)
a) \(10\)
b) \(6\)
c) \(2\)
d) \(0\)
e) \(-2\)
\(\log \left ( \frac{x^{2}+16}{x} \right )=1\\\log \left ( \frac{x^{2}+16}{x} \right )=\log 10\rightarrow \frac{x^{2}+16}{x}=10\textrm{ atau}\\x^{2}-10x+16=0 \textrm{ berarti:}x_{1}+x_{2}=-\left ( -\frac{10}{1} \right )=10\)
28. \(\textrm{Jika }^{2}\log a=,\textrm{ maka } \left [ \left ( a^{2} \right )^{3} \right ]^{-\frac{1}{2}}=\)
a) \(\frac{1}{64}\)
b) \(\frac{1}{81}\)
c) \(\frac{1}{729}\)
d) \(\frac{1}{512}\)
e) \(\frac{1}{4096}\)
\(^{2}\log a=3\rightarrow a=2^{3}=8\\\left [ \left ( a^{2} \right )^{3} \right ]^{-\frac{1}{2}}=a^{\left ( 2\times 3\times \left ( -\frac{1}{2} \right ) \right )}=a^{-3}=8^{-3}=\frac{1}{512}\)
29. \(\textrm{Jika }a> 1,\textrm{ maka penyelesaian }\left ( ^{a}\log \left ( 2x+1 \right )\left ( ^{3}\log \sqrt{a} \right ) \right )=1\textrm{ adalah...}\)
a) \(1\)
b) \(2\)
c) \(3\)
d) \(4\)
e) \(5\)
\(^{a}\log \left ( 2x+1 \right )\times ^{3}\log \left ( a \right )^{\frac{1}{2}}=1\\\frac{1}{2}\times ^{3}\log a\times ^{a}\log \left ( 2x+1 \right )=1\\^{3}\log \left ( 2x+1 \right )=2\times ^{3}\log 3\rightarrow 2x+1=9\Rightarrow x=4\)
30. \(\textrm{Jika x memenuhi persamaan }^{4}\log x-^{4}\log ^{4}\log 16 =2, \textrm{ maka }^{16}\log x =...\)
a) \(4\)
b) \(2\)
c) \(1\)
d) \(-2\)
e) \(-4\)
\(^{4}\log ^{4}\log x-^{4}\log ^{4}\log \left ( ^{4}\log 16 \right )=2\\^{4}\log x\left ( \frac{^{4}\log x}{^{4}\log \left ( 2 \right )} \right )= ^{4}\log 16\rightarrow ^{2}\log x=16\\\textrm{Maka dapat diperoleh :}^{16}\log x=^{2^{4}}\log x=\frac{1}{4}\times ^{2}\log x=\frac{1}{4}\times 16=4\)
