Persamaan Logaritma 4

 D. Persamaan Logaritma Bentuk  $^{h(x)}\log f(x)=\: ^{h(x)}\log g(x)$.

Syarat penyelesaian dari bentuk:

$\begin{aligned}&\textrm{Jika}\: \: ^{h(x)}\log f(x)=\, ^{h(x)}\log g(x)\\ &\textrm{dengan}\: \: f(x)\: \: \textrm{dan}\: \: g(x)\: \: \textrm{keduanya positif}\\ &\textrm{serta}\: \: h(x)>0,\: \: \textrm{dan}\: \: h(x)\neq 1,\\ &\textrm{maka}\: \: f(x)=g(x) \end{aligned}$.

$\begin{aligned}&\textbf{atau}\\ &\textrm{Pernyataan}\: \: \color{red}^{h(x)}\log f(x)=\, ^{h(x)}\log g(x)\\ &\textrm{akan bernilai benar jika}\\ &(1)\quad \color{blue}h(x)>0,\: \: h(x)\neq 1\\ &(2)\quad \color{blue}f(x)>0,\: g(x)>0\\ &(3)\quad \color{blue}f(x)=g(x)\end{aligned}$.

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

$\begin{array}{ll}\\ 1.&\textrm{Tentukan himpunan penyelesaian dari}\\ &\textrm{a}.\quad ^{x}\log (2x-3)= \, ^{x}\log (x-1)\\ &\textrm{b}.\quad ^{x}\log (2x^{2}+11x-6)=\, ^{x}\log (x^{2}+10x)\\ &\textrm{c}.\quad ^{x}\log (x-1)+\displaystyle \frac{1}{^{x+6}\log x}=2+\, \displaystyle \frac{1}{^{2}\log x}\\ &\textrm{d}.\quad ^{2x-1}\log (x^{3}+3x^{2}-4x-1)=\, ^{2x-1}\log (2x^{2}+3)\\\\ &\textrm{Jawab}:\\ &\begin{aligned}\textbf{a}.\: \: \: \textrm{Dik}&\textrm{etahui}\\ &^{x}\log (2x-3)=\, ^{x}\log (x-1)\\ (\ast )\: \: &\color{blue}\textrm{Syarat numerus dan bilangan pokok}\\ &\begin{array}{|l|l|}\hline \: \: \: \: \qquad\textrm{Syarat}&\: \: \quad\textrm{hasil}\\\hline h(x)>0, \: h(x)\neq 1&x>0\\\hline f(x)>0&\begin{aligned}2x&-3>0\\ \Leftrightarrow \: \: &2x>3\\ \Leftrightarrow \: \: &x>\displaystyle \frac{3}{2} \end{aligned}\\\hline g(x)>0&\begin{aligned}x-&1>0\\ \Leftrightarrow \: \: &x>1 \end{aligned}\\\hline \end{array} \\ &\textrm{Syarat numerusnya},\: \: \color{red}x>\displaystyle \frac{3}{2}\\ (\ast )\: \: &\color{blue}\textrm{Syarat kedua},\: \: \color{black}f(x)=g(x)\\ &\Leftrightarrow \quad 2x-3=x-1\\ &\Leftrightarrow \quad 2x-x=3-1\\ &\Leftrightarrow \quad x=2\\ &\textrm{Karena}\: \: x>\displaystyle \frac{3}{2},\\ & \textrm{maka nilai}\: \: x=2\: \: \textrm{memenuhi}\\ (\ast )\: \: &\textrm{Jadi},\: \: \textrm{HP}=\color{red}\left \{ 2 \right \} \end{aligned}\\ &\begin{aligned}\textbf{b}.\: \: \: \textrm{Dik}&\textrm{etahui}\\ &^{x}\log (2x^{2}+11x-6)=\, ^{x}\log (x^{2}+10)\\ (\ast )\: \: &\color{blue}\textrm{Syarat numerus dan bilangan pokok}\\ &\begin{array}{|l|l|}\hline \: \: \: \: \qquad\textrm{Syarat}&\: \: \quad\textrm{hasil}\\\hline h(x)>0,\: h(x)\neq 1&x>0\\\hline f(x)>0&\begin{aligned}2x^{2}&+11x-6>0\\ \Leftrightarrow \: \: &(x+6)(2x-1)>0\\ \Leftrightarrow \: \: &x<-6\: \: \textrm{atau}\: \: x>\displaystyle \frac{1}{2} \end{aligned}\\\hline g(x)>0&\begin{aligned}x^{2}&+10x>0\\ \Leftrightarrow \: \: &x(x+10)>0\\ \Leftrightarrow \: \: &x<-10\: \: \textrm{atau}\: \: x>0 \end{aligned}\\\hline \end{array} \\ &\textrm{Syarat numerusnya},\: \: \color{red}x>\displaystyle \frac{1}{2}\\ (\ast )\: \: &\color{blue}\textrm{Syarat kedua},\: \: \color{black}f(x)=g(x)\\ &\Leftrightarrow \quad 2x^{2}+11x-6=x^{2}+10x\\ &\Leftrightarrow \quad x^{2}+x-6=0\\ &\Leftrightarrow \quad (x+3)(x-2)=0\\ &\Leftrightarrow \quad x=-3\: \: \textrm{atau}\: \: x=2\\ &\textrm{Karena}\: \: x>\displaystyle \frac{1}{2},\\ & \textrm{maka nilai}\: \: x=2\: \: \textrm{memenuhi}\\ (\ast )\: \: &\textrm{Jadi},\: \: \textrm{HP}=\color{red}\left \{ 2 \right \} \end{aligned}\\ &\begin{aligned}\textbf{c}.\: \: \: \textrm{Dik}&\textrm{etahui}\\ &^{x}\log (x-1)+\displaystyle \frac{1}{^{x+6}\log x}=2+\, \displaystyle \frac{1}{^{2}\log x}\\ &\Leftrightarrow \: ^{x}\log (x-1)+\, ^{x}\log (x+6)=\, ^{x}\log x^{2}+\, ^{x}\log 2\\ &\Leftrightarrow \: ^{x}\log (x-1)(x+6)=\, ^{x}\log 2x^{2}\\ &\Leftrightarrow \: ^{x}\log x^{2}+5x-6=\, ^{x}\log 2x^{2}\\ (\ast )\: \: &\color{blue}\textrm{Syarat numerus dan bilangan pokok}\\ &\begin{array}{|l|l|}\hline \: \: \: \: \qquad\textrm{Syarat}&\: \: \quad\textrm{hasil}\\\hline h(x)>0,\: h(x)\neq 1&x>0\\\hline f(x)>0&\begin{aligned}x^{2}&+5x-6>0\\ \Leftrightarrow \: \: &(x+6)(x-1)>0\\ \Leftrightarrow \: \: &x<-6\: \: \textrm{atau}\: \: x>1 \end{aligned}\\\hline g(x)>0&\begin{aligned}2x^{2}&>0\\ \Leftrightarrow \: \: &x>0 \end{aligned}\\\hline \end{array} \\ &\textrm{Syarat numerusnya},\: \: \color{red}x>\displaystyle 1\\ (\ast )\: \: &\color{blue}\textrm{Syarat kedua},\: \: \color{black}f(x)=g(x)\\ &\Leftrightarrow \quad x^{2}+5x-6=2x^{2}\\ &\Leftrightarrow \quad x^{2}-5x+6=0\\ &\Leftrightarrow \quad (x-2)(x-3)=0\\ &\Leftrightarrow \quad x=2\: \: \textrm{atau}\: \: x=3\\ &\textrm{Karena}\: \: x>\displaystyle 1,\\ & \textrm{maka nilai}\: \: x=2\: \: \textrm{dan}\: \: x=3\: \: \textrm{memenuhi}\\ (\ast )\: \: &\textrm{Jadi},\: \: \textrm{HP}=\color{red}\left \{ 2,3 \right \} \end{aligned}\\ &\begin{aligned}\textbf{d}.\: \: \: \textrm{Dik}&\textrm{etahui}\\ &^{2x-1}\log (x^{3}+3x^{2}-4x-1)=\, ^{2x-1}\log (2x^{2}+3)\\ (\ast )\: \: &\color{blue}\textrm{Syarat numerus dan bilangan pokok}\\ &\begin{array}{|l|l|}\hline \: \: \: \: \qquad\textrm{Syarat}&\qquad \: \: \quad\textrm{hasil}\\\hline h(x)>0,\: h(x)\neq 1&2x-1>0\Leftrightarrow x>\displaystyle \frac{1}{2}\\\hline f(x)>0&\begin{aligned}x^{3}&+3x^{2}-4x-1>0\\ \bullet \: \: &\textrm{Susah difaktorkan}\\ \bullet \: \: &\textrm{gunakan uji nilai} \end{aligned}\\\hline g(x)>0&\begin{aligned}2x^{2}&+3>0\\ \bullet \: \: &a>0,\: D<0\\ \bullet \: \: &\color{red}\textbf{Definit positif} \end{aligned}\\\hline \end{array} \\ &\textrm{Syarat basis/bilangan pokoknya},\: \: \color{red}x>\displaystyle \frac{1}{2}\\ (\ast )\: \: &\color{blue}\textrm{Syarat kedua},\: \: \color{black}f(x)=g(x)\\ &\Leftrightarrow \quad x^{3}+3x^{2}-4x-1=2x^{2}+3\\ &\Leftrightarrow \quad x^{3}+x^{2}-4x-4=0\\ &\Leftrightarrow \quad x^{2}(x+1)-4(x+1)=0\\ &\Leftrightarrow \quad (x+1)(x^{2}-4)=0\\ &\Leftrightarrow \quad (x+1)(x+2)(x-2)=0\\ &\Leftrightarrow \quad x=-1\: \: \textrm{atau}\: \: x=-2\: \: \textrm{atau}\: \: x=2\\ &\textrm{Karena basisnya}\: \: x>\displaystyle \frac{1}{2},\\ &\textrm{maka nilai yang memenuhi hanya}\: \: x=2\: \: \textrm{saja}\\ &\textrm{dan nilai untuk numerusnya juga memenuhi}\\ &\textrm{yaitu}:\: (2)^{3}+3(2)^{2}-4(2)-1=11>0\\ &\textrm{demikian pula untuk}:\: 2(2)^{3}+3=19>0\\ (\ast )\: \: &\textrm{Jadi},\: \: \textrm{HP}=\color{red}\left \{ 2 \right \} \end{aligned} \end{array}$.

Penjelasan untuk jawaban 1. d  tentang definit positif  di sini dan di sini

$\LARGE\colorbox{aqua}{LATIHAN SOAL}$.

$\begin{array}{ll}\\ 2.&\textrm{Tentukan himpunan penyelesaian dari}\\ &\textrm{a}.\quad ^{x}\log (2x+3)= \, ^{x}\log (x+7)\\ &\textrm{b}.\quad ^{x}\log (x+12)- \, ^{x}\log (4x+1)=0\\ &\textrm{c}.\quad ^{x-2}\log (x^{2}-3)=\, ^{x-2}\log x\\ &\textrm{d}.\quad ^{3x-2}\log (x^{2}-2x+4)=\, ^{3x-2}\log (5-4x)\\\\ \end{array}$