Persamaan Logaritma

Persamaan logaritma adalah persamaan untuk numerus atau bilangan pokok logaritmanya mengandung variabel x

\color{blue}\begin{array}{|c|l|l|}\hline \textrm{No}.&\textrm{Bentuk}&\textrm{Persamaan}\\\hline 1.&^{a}\log f(x)=\, ^{a}\log p&f(x)=p,\quad \textrm{syarat}\: \: f(x)> 0\\\hline 2.&^{a}\log f(x)=\, ^{b}\log f(x)&f(x)=1,\quad \textrm{dengan}\: \: a\neq b\\\hline 3.&^{a}\log f(x)=\, ^{a}\log g(x)&f(x)=g(x),\: \textrm{keduanya positif}\\\hline 4.&^{h(x)}\log f(x)=\, ^{h(x)}\log g(x)&f(x)=g(x),\: \textrm{keduanya positif}\\\hline &&\qquad h(x)> 0,\: \: \&\: h(x)\neq 1\\\hline 5.&A\left ( ^{a}\log x \right )^{2}+B\left ( ^{a}\log x \right )+C=0&a\neq 1,\: A\neq 0,\: A,B,C\in \mathbb{R}\\\hline \end{array}.

\colorbox{yellow}{\LARGE\fbox{CONTOH SOAL}}.

\color{blue}\begin{array}{|c|l|}\hline \textrm{Bentuk}&\textrm{Persamaan}\\\hline ^{a}\log f(x)=\, ^{a}\log p&f(x)=p,\quad \textrm{syarat}\: \: f(x)> 0\\\hline \textrm{Contoh}&\begin{aligned}\textrm{Diketahui persamaan}&\: \textrm{berikut}\\ ^{3}\log (2x-1)&=2\\ \textrm{Tentukanlah penyele}&\textrm{saiannya}\\ \textrm{Jawab}:\qquad\qquad\quad\: \: \: &\\ ^{3}\log (2x-1)&=2\\ ^{3}\log (2x-1)&=\, ^{3}\log 3^{2}\\ 2x-1&=9\\ 2x&=10\\ x&=5\\ \textrm{atau boleh juga}\qquad& \\ ^{3}\log (2x-1)&=2\\ 2x-1&=3^{2}\\ 2x-1&=9\\ 2x&=10\\ x&=5 \end{aligned} \\\hline \end{array}.

\color{blue}\begin{array}{|c|l|}\hline \textrm{Bentuk}&\qquad\qquad\textrm{Persamaan}\\\hline ^{a}\log f(x)=\: ^{a}\log p &f(x)=p,\quad \textrm{syarat}\: \: f(x)> 0\\\hline \textrm{Contoh}&\begin{aligned}\textrm{Diketahui persamaan}&\: \textrm{berikut}\\ ^{2}\log (x^{2}-4x+5)&=1\\ \textrm{Tentukanlah penyele}&\textrm{saiannya}\\ \textrm{Jawab}:\qquad\qquad\quad\: \: \: &\\ ^{2}\log (x^{2}-4x+5)&=1\\ ^{2}\log (x^{2}-4x+5)&=^{2}\log 2^{1}\\ x^{2}-4x+5&=2\\ x^{2}-4x+3&=0\\ (x-1)(x-3)&=0\\ x=1\: \: v\: \: x&=3 \end{aligned} \\\hline \end{array}.

\color{green}\begin{array}{|c|l|}\hline \textrm{Bentuk}&\qquad\qquad\quad\textrm{Persamaan}\\\hline ^{a}\log f(x)=\: ^{b}\log f(x) &f(x)=1,\quad \textrm{dengan}\: \: a\neq b\\\hline \textrm{Contoh}&\begin{aligned}\textrm{Diketahui persamaan}&\: \textrm{berikut}\\ ^{2}\log (2x-5)&=\, ^{3}\log (2x-5)\\ \textrm{Tentukanlah penyele}&\textrm{saiannya}\\ \textrm{Jawab}:\qquad\qquad\quad\: \: \: &\\ ^{2}\log (2x-5)&=\, ^{3}\log (2x-5)\\ 2x-5&=1\\ 2x&=6\\ x&=3 \end{aligned} \\\hline \end{array}.

\color{green}\begin{array}{|c|l|}\hline \textrm{Bentuk}&\qquad\qquad\qquad\textrm{Persamaan}\\\hline ^{a}\log f(x)=\: ^{b}\log f(x) &f(x)=1,\quad \textrm{dengan}\: \: a\neq b\\\hline \textrm{Contoh}&\begin{aligned}\textrm{Diketahui persamaan}&\: \textrm{berikut}\\ ^{2}\log (x^{2}-x+1)&=\, ^{2}\log (x^{2}-x+1)\\ \textrm{Tentukanlah penyele}&\textrm{saiannya}\\ \textrm{Jawab}:\qquad\qquad\quad\: \: \: &\\ ^{2}\log (x^{2}-x+1)&=\,^{2}\log (x^{2}-x+1)\\ (x^{2}-x+1)&=1\\ x^{2}-x&=0\\ x(x-1)&=0\\ x=0\: \: v\: \: x&=1 \end{aligned} \\\hline \end{array}.

\color{red}\begin{array}{|c|l|}\hline \textrm{Bentuk}&\qquad\qquad\qquad\qquad\qquad \textrm{Persamaan}\\\hline ^{a}\log f(x)=\: ^{a}\log g(x) &f(x)=g(x),\quad \textrm{asalkan}\: \: f(x)\&g(x)\: \textbf{positif}\\\hline \textrm{Contoh}&\begin{aligned}\textrm{Diketahui persamaan}&\: \textrm{berikut}\\ \log (x^{2}-4x+2)&=\log (x+2)\\ \textrm{Tentukanlah penyele}&\textrm{saiannya}\\ \textrm{Jawab}:\qquad\qquad\quad\: \: \: &\\ \log (x^{2}-4x+2)&=\log (x+2)\\ (x^{2}-4x+2)&=(x+2)\\ x^{2}-5x&=0\\ x(x-5)&=0\\ x=0\: \: \textrm{v}\: \: x&=5\\ \textbf{syarat}\qquad x&=0\begin{cases} f(0) &=0^{2}-4.0+2=2\: (\textbf{positif}) \\ g(0) &=0+2=2\: (\textbf{positif}) \end{cases}\\ \textrm{dan}\qquad x&=5\begin{cases} f(5) &=5^{2}-4.5+2=7\: (\textbf{positif}) \\ g(5) &=5+2=7\: (\textbf{positif}) \end{cases}\\ \therefore \quad \textrm{keduanya}&\: \textrm{memenuhi} \end{aligned} \\\hline \end{array}.

\color{red}\begin{array}{|c|l|}\hline \textrm{Bentuk}&\qquad\qquad\qquad\qquad\qquad\qquad \textrm{Persamaan}\\\hline ^{a}\log f(x)=\: ^{a}\log g(x) &f(x)=g(x),\quad \textrm{asalkan}\: \: f(x)\&g(x)\: \textbf{positif}\\\hline \textrm{Contoh}&\begin{aligned}\textrm{Diketahui persamaan}&\: \textrm{berikut}\\ \log (x^{2}+5x-7)&=\log (x-2)\\ \textrm{Tentukanlah penyele}&\textrm{saiannya}\\ \textrm{Jawab}:\qquad\qquad\quad\: \: \: &\\ \log (x^{2}+5x-7)&=\log (x-2)\\ (x^{2}+5x-7)&=(x-2)\\ x^{2}+4x-5&=0\\ (x+5)(x-1)&=0\\ x=-5\: \: \textrm{v}\: \: x&=1\\ \textbf{syarat}\qquad x=-5&\begin{cases} f(-5) &=(-5)^{2}+5(-5)-7=-7\: (\textbf{negatif}) \\ g(-5) &=-5-2=-7\: (\textbf{negatif}) \end{cases}\\ \textrm{dan}\qquad x=1&\begin{cases} f(1) &=1^{2}+5.1-7=-1\: (\textbf{negatif}) \\ g(1) &=1-2=-1\: (\textbf{negatif}) \end{cases}\\ \therefore \quad \textrm{keduanya}&\: \textbf{tidak memenuhi} \end{aligned} \\\hline \end{array}.

\color{blue}\begin{array}{|c|l|}\hline \textrm{Bentuk}&\qquad\qquad\qquad\qquad \textrm{Persamaan}\\\hline ^{h(x)}\log f(x)=\: ^{h(x)}\log g(x) &f(x)=g(x),\quad \textrm{asalkan}\: \: f(x)\&g(x)\: \textbf{positif}\\ &\textrm{serta}\: \: h(x)> 1\: \: \textrm{dan}\: \: h(x)\neq 1\\\hline \textrm{Contoh}&\begin{aligned}\textrm{Diketahui persamaan}&\: \textrm{berikut}\\ ^{x}\log (x+1)&=\, ^{x}\log (2x-1)\\ \textrm{Tentukanlah penyele}&\textrm{saiannya}\\ \textrm{Jawab}:\qquad\qquad\quad\: \: \: &\\ ^{x}\log (x+1)&=\, ^{x}\log (2x-1)\\ x+1&=2x-1\\ x&=2\\ \textbf{syarat}\qquad x=2&\begin{cases} f(2) &=2+1=3\: (\textbf{positif}) \\ g(2) &=2.2-1=3\: (\textbf{positif}) \end{cases}\\ \textrm{dan}\qquad &x=2> 0,\: \: \textrm{serta}\: \: x\neq 1\\ \therefore \quad \textrm{maka}\: \: x=2&\: \: \textbf{memenuhi} \end{aligned} \\\hline \end{array}.

\color{blue}\begin{array}{|c|l|}\hline \textrm{Bentuk}&\qquad\qquad\qquad\qquad \textrm{Persamaan}\\\hline ^{h(x)}\log f(x)=\: ^{h(x)}\log g(x) &f(x)=g(x),\quad \textrm{asalkan}\: \: f(x)\&g(x)\: \textbf{positif}\\ &\textrm{serta}\: \: h(x)> 1\: \: \textrm{dan}\: \: h(x)\neq 1\\\hline \textrm{Contoh}&\begin{aligned}\textrm{Diketahui persamaan}&\: \textrm{berikut}\\ ^{2x-5}\log (2x+1)&=\, ^{2x-5}\log (x+4)\\ \textrm{Tentukanlah penyele}&\textrm{saiannya}\\ \textrm{Jawab}:\qquad\qquad\quad\: \: \: &\\ ^{2x-5}\log (2x+1)&=\, ^{2x-5}\log (x+4)\\ 2x+1&=x+4\\ x&=3\\ \textbf{syarat}\qquad x=3&\begin{cases} f(3) &=2(3)+1=7\: (\textbf{positif}) \\ g(3) &=3+4=7\: (\textbf{positif}) \end{cases}\\ \textrm{dan}\qquad x&=3\quad \textrm{saat}\:\: 2(3)-5=1> 0\\ &\textrm{tetapi sayangnya sama dengan 1}\\ \therefore \quad \textrm{maka}\: \: x=3&\: \: \textbf{tidak memenuhi} \end{aligned} \\\hline \end{array}.

\color{blue}\begin{array}{|c|l|}\hline \textrm{Bentuk}&\qquad\qquad\qquad\qquad\qquad\qquad \textrm{Persamaan}\\\hline A\left ( ^{a}\log x \right )^{2}+B\left ( ^{a}\log x \right )+C=0 &a> 0,\: a\neq 1,\quad A,B,C\in \mathbb{R}\\ &\textrm{serta}\: \: A\neq 0\\\hline \textrm{Contoh}&\begin{aligned}\textrm{Diketahui persamaan}&\: \textrm{berikut}\\ ^{2}\log^{2}x-2.\, ^{2}\log x&-3=0\\ \textrm{Tentukanlah penyele}&\textrm{saiannya}\\ \textrm{Jawab}:\qquad\qquad\quad\: \: \: &\\ ^{2}\log^{2}x-2.\, ^{2}\log x&-3=0\\ \textrm{misalkan}\quad y&=\, ^{2}\log x,\quad \textrm{maka}\\ y^{2}-2y-3&=0\\ (y-3)(y+1)&=0\\ \textrm{didapatkan}&\: \: y=3\Rightarrow \, ^{2}\log x=3\Rightarrow x=2^{3}=8\\ &\: \: y=-1\Rightarrow \, ^{2}\log x=-1\Rightarrow x=2^{-1}\\ &\: \: \, =\displaystyle \frac{1}{2}\\ \therefore \qquad \textrm{penyelesaiaanya}&\: \textrm{adalah}\: \: x=\displaystyle \frac{1}{2}\: \: \textrm{dan}\: \: x=8 \end{aligned} \\\hline \end{array}.

\color{blue}\begin{array}{|c|l|}\hline \textrm{Bentuk}&\qquad\qquad\qquad\qquad\qquad\qquad \textrm{Syarat}\\\hline A\left ( ^{a}\log x \right )^{2}+B\left ( ^{a}\log x \right )+C=0 &a> 0,\: a\neq 1,\quad A,B,C\in \mathbb{R}\\ &\textrm{serta}\: \: A\neq 0\\\hline \textrm{Contoh}&\begin{aligned}\textrm{Diketahui persamaan}&\: \textrm{berikut}\\ ^{2}\log x^{^{1+\, ^{2}\log x}}&=2\\ \textrm{Tentukanlah penyele}&\textrm{saiannya}\\ \textrm{Jawab}:\qquad\qquad\quad\: \: \: &\\ ^{2}\log x^{^{1+\, ^{2}\log x}}&=2\\ \left ( 1+\, ^{2}\log x \right ).^{2}\log x&=2\\ ^{2}\log x+\, ^{2}\log ^{2}x&-2=0\\ \textrm{misalkan}\quad y&=\, ^{2}\log x,\quad \textrm{maka}\\ y^{2}+y-2&=0\\ (y+2)(y-1)&=0\\ y=-2\: \: \textrm{v}\: \: y&=1\\ \textrm{didapatkan}&\: \: y=-2\Rightarrow \, ^{2}\log x=-2\\ &\: \: \: x=2^{-2}=\displaystyle \frac{1}{4}\\ &\: \: y=1\Rightarrow \, ^{2}\log x=1\Rightarrow x=2^{1}=2\\ \therefore \textrm{Penyelesaiannya}&\: \textrm{adalah}\: \: x=\displaystyle \frac{1}{4}\: \: \textrm{dan}\: \: x=2 \end{aligned} \\\hline \end{array}.

\colorbox{magenta}{\LARGE\fbox{LATIHAN SOAL}}.

\color{blue}\begin{array}{ll}\\ 1.&\textrm{Tentukanlah penyelesaian setiap persamaan berikut}\\ &\textrm{a}.\quad \log x=-\log 2\qquad\qquad\qquad\qquad \textrm{f}.\quad \log x^{2}=\log 4+\log (x+3)\\ &\textrm{b}.\quad \log x=\frac{1}{2}.\log 8\qquad\qquad\qquad\: \: \: \: \quad \textrm{g}.\quad ^{3}\log x^{2}=\, ^{3}\log 8x-1\\ &\textrm{c}.\quad \log x^{2}=\log x\qquad\qquad\qquad\: \: \qquad \textrm{h}.\quad ^{2}\log x^{2}=1+\, ^{2}\log (x+2)\\ &\textrm{d}.\quad \log (x+2)=\log x+\log 2\qquad\: \, \quad \textrm{i}.\quad \log x=\left ( \log \log 5\sqrt{5} \right )+1\\ &\textrm{e}.\quad \log x-\log 2=\log (x-2)\qquad\: \, \quad \textrm{j}.\quad \log \log x=1+\log 2 \end{array}.

\color{blue}\begin{array}{ll}\\ 2.&\textrm{Tentukanlah penyelesaian setiap persamaan berikut}\\ &\textrm{a}.\quad ^{2}\log (x-4)+\, ^{2}\log (x-6)=3\\ &\textrm{b}.\quad ^{2}\log (x-5)+\, ^{2}\log (x-2)=\, ^{9}\log 81\\ &\textrm{c}.\quad ^{2}\log (x-2)+\, ^{2}\log (x-3)=\, ^{^{\frac{1}{3}}}\log 2\times \, ^{2}\log \displaystyle \frac{1}{3}\\ &\textrm{d}.\quad ^{0,25}\log (x-4)+\, ^{16}\log (x+2)=0\\ &\textrm{e}.\quad ^{2}\log (2x-1)-\, ^{4}\log \left ( 3x-\displaystyle \frac{3}{2} \right )=1 \end{array}.

\color{red}\begin{array}{ll}\\ 3.&\textrm{Tentukanlah penyelesaian setiap persamaan berikut}\\ &\textrm{a}.\quad \log \left ( \log (3x+4)+2 \right )=\log 4\\ &\textrm{b}.\quad \log \left ( \log (x-3)+\log 2 \right )=\log \log 4x\\ &\textrm{c}.\quad ^{7}\log \left ( \log x^{5}+14 \right )=\, ^{7}\log \left ( \log \displaystyle \frac{x^{2}}{10} \right )\\ &\textrm{d}.\quad ^{2}\log \left ( ^{2}\log \left ( 2^{x+1}+8 \right ) \right )=1+\: ^{2}\log x\\ &\textrm{e}.\quad ^{4}\log \, ^{3}\log \, ^{2}\log x=0\\ &\textrm{f}.\quad ^{4}\log \left ( 2.^{3}\log \left ( 1+\, ^{2}\log \left ( 1+3.^{2}\log x \right ) \right ) \right )=\displaystyle \frac{1}{2}\\ &\textrm{g}.\quad ^{a}\log \left ( 1+\, ^{b}\log \left ( 1+\, ^{c}\log \left ( 1+\, ^{p}\log x \right ) \right ) \right )=0 \end{array}.

\color{blue}\begin{array}{ll}\\ 4.&\textrm{Tentukanlah penyelesaian setiap persamaan berikut}\\ &\textrm{a}.\quad \displaystyle \frac{1}{^{2x-1}\log x}+\frac{1}{^{x+6}\log x}=\displaystyle \frac{1}{^{x+10}\log x}+1\\ &\textrm{b}.\quad ^{x+13}\log (x-1)-\, ^{\frac{1}{x+13}}\log 8=\displaystyle \frac{\log^{2}5-\log ^{2}2 }{\log 2,5}\\ &\textrm{c}.\quad \displaystyle \frac{1}{^{x-2}\log x}+\, ^{x}\log (x-3)+\, ^{x}\log 4=\displaystyle \frac{1}{^{8}\log x}\\ &\textrm{d}.\quad \displaystyle \frac{1}{^{x+6}\log x}+\, ^{x}\log (x-1)=2+\displaystyle \frac{1}{^{2}\log x}\\ \end{array}.

\color{blue}\begin{array}{ll}\\ 5.&\textrm{Tentukanlah penyelesaian setiap persamaan berikut}\\ &\textrm{a}.\quad ^{5}\log ^{2}x-\, ^{5}\log x^{4}+\, ^{5}\log 125=0\\ &\textrm{b}.\quad ^{5}\log ^{2}x+5^{.^{^{5}}\log 30-\, ^{^{5}}\log 3}=\, ^{5}\log x^{6}+25^{.^{^{5}}\log \sqrt{5}}\\ &\textrm{c}.\quad x^{.^{^{2}}\log x}=\displaystyle \frac{x^{4}}{8}\\ &\textrm{d}.\quad ^{2}\log x^{1+\, ^{2}\log x}=6\\ \end{array}.

DAFTAR PUSTAKA

  1. Budhi, W.S., Widodo, U. 2017. Matematika untuk SMA/MA Kelas X Kelompok Peminatan Matematika dan Ilmu-Ilmu Alam. Jakarta: ERLANGGA.
  2. Wirodikromo, S. 1996. Matematika untuk SMU Kelas 2 Caturwulan 3. Jakarta: ERLANGGA.

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