Contoh Soal 7 Fungsi Logaritma (Pertidaksamaan Logaritma)

 $\begin{array}{ll}\\ 32.&\textrm{Agar}\: \: \log \left ( x^{2}-1 \right )<0\: \: \textrm{maka}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&-1<x<1\\ \textrm{b}.&-\sqrt{2}<x<\sqrt{2}\\ \textrm{c}.&x<-1\: \: \textrm{atau}\: \: x>1\\ \textrm{d}.&x<-\sqrt{2}\: \: \textrm{atau}\: \: x>\sqrt{2}\\ \color{red}\textrm{e}.&-\sqrt{2}<x<-1\: \: \textrm{atau}\: \: 1<x<\sqrt{2} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{e}\\ &\color{blue}\begin{aligned}&\log \left ( x^{2}-1 \right )<0\\ &\textrm{Diketahui}\: \: \color{red}\log f(x)<0,\: \: \color{blue}\textrm{maka}\\ &\color{purple}\textrm{Syarat (1)},\: \: \color{red}f(x)>0\\ &\Leftrightarrow x^{2}-1>0\\ &\Leftrightarrow x<-1\: \: \textrm{atau}\: \: x>1\\ &\color{purple}\textrm{Syarat (2)},\: \: \log \left ( x^{2}-1 \right )<0\\ &\log \left ( x^{2}-1 \right )<\log 1\\ &\Leftrightarrow x^{2}-1<1\\ &\Leftrightarrow x^{2}-2<0\\ &\Leftrightarrow x^{2}-\left ( \sqrt{2} \right )^{2}<0\\ &\Leftrightarrow -\sqrt{2}<x<\sqrt{2}\\ &\textrm{Jadi},\: \: \color{red}-\sqrt{2}<x<-1\: \: \textrm{atau}\: \: 1<x<\sqrt{2} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 33.&\textrm{Himpunan penyelesaian dari}\\ &^{.^{\frac{1}{2}}}\log \left ( x^{2}-3 \right )>0 \: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&\left \{ x|-2<x<-\sqrt{3}\: \: \textrm{atau}\: \: \sqrt{3}<x<2 \right \}\\ \textrm{b}.&\left \{ x|-\sqrt{3}<x<-1\: \: \textrm{atau}\: \: \sqrt{3}<x<2 \right \}\\ \textrm{c}.&\left \{ x|-2<x<-\sqrt{3} \right \}\\ \textrm{d}.&\left \{ x|-2<x<-\sqrt{3} \right \}\\ \textrm{e}.&\left \{ x|\sqrt{3}<x<2 \right \} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}&^{.^{\frac{1}{2}}}\log \left ( x^{2}-3 \right )>0\\ &\textrm{Diketahui}\: \: \color{red}^{.^{\frac{1}{2}}}\log f(x)>0,\: \: \color{blue}\textrm{maka}\\ &\color{purple}\textrm{Syarat (1)},\: \: \color{red}f(x)>0\\ &\Leftrightarrow x^{2}-3>0\\ &\Leftrightarrow x<-\sqrt{3}\: \: \textrm{atau}\: \: x>\sqrt{3}\\ &\color{purple}\textrm{Syarat (2)},\: \: ^{.^{\frac{1}{2}}}\log \left ( x^{2}-3 \right )>0\\ &^{.^{\frac{1}{2}}}\log \left ( x^{2}-3 \right )>\: ^{.^{\frac{1}{2}}}\log 1\\ &\Leftrightarrow x^{2}-3<1\quad \left (\color{black}\textrm{karena basisnya}\: \: \displaystyle \frac{1}{2}<1 \right )\\ &\Leftrightarrow x^{2}-4<0\\ &\Leftrightarrow x^{2}-2^{2}>0\\ &\Leftrightarrow -2<x<2\\ &\textrm{Jadi},\: \: \color{red}-2<x<-\sqrt{3}\: \: \textrm{atau}\: \: \sqrt{3}<x<2 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 34.&\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi}\\ &^{2}\log \left ( x^{2}-x \right )\leq 1 \: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&x<0\: \: \textrm{atau}\: \: x>1\\ \textrm{b}.&-1\leq x\leq 2,\: x\neq 1\: \: \textrm{atau}\: \: x\neq 0\\ \color{red}\textrm{c}.&-1\leq x< 0\: \: \textrm{atau}\: \: 1<x\leq 2\\ \textrm{d}.&-1< x\leq 0\: \: \textrm{atau}\: \: 1\leq x< 2\\ \textrm{e}.&-1\leq x\leq 0\: \: \textrm{atau}\: \: 1\leq x\leq 2 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\color{blue}\begin{aligned}&^{2}\log \left ( x^{2}-x \right )\leq 1\\ &\textrm{Diketahui}\: \: \color{red}^{2}\log f(x)>0,\: \: \color{blue}\textrm{maka}\\ &\color{purple}\textrm{Syarat (1)},\: \: \color{red}f(x)>0\\ &\Leftrightarrow x^{2}-x>0\Leftrightarrow x(x-1)>0\\ &\Leftrightarrow x<0\: \: \textrm{atau}\: \: x>1\\ &\color{purple}\textrm{Syarat (2)},\: \: ^{2}\log \left ( x^{2}-x \right )\leq 1\\ &^{2}\log \left ( x^{2}-x \right )\leq \: ^{2}\log 2\\ &\Leftrightarrow x^{2}-x\leq 2\\ &\Leftrightarrow x^{2}-x-2\leq 0\\ &\Leftrightarrow (x+1)(x-2)\leq 0\\ &\Leftrightarrow -1\leq x\leq 2\\ &\textrm{Jadi},\: \: \color{red}-1\leq x< 0\: \: \textrm{atau}\: \: 1<x\leq 2 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 35.&\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi}\\ &\left | \log (x+1) \right |> 1 \: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&x<-0,9\: \: \textrm{atau}\: \: x>9\\ \textrm{b}.&x<-9\: \: \textrm{atau}\: \: x>9\\ \color{red}\textrm{c}.&-1<x<-0,9\: \: \textrm{atau}\: \: x>9\\ \textrm{d}.&-9< x<0,9\\ \textrm{e}.&-0,9<x<9 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\color{blue}\begin{aligned}&\textrm{Ingat bahwa}\\ &\left | x \right |>A\Leftrightarrow \color{black}x<-A\: \: \textrm{atau}\: \: x>A,\: \: \color{red}A>0\\ &\Leftrightarrow \log (x+1)<-1\: \: \textrm{atau}\: \: \log (x+1)>1\\ &\color{red}\textrm{Syarat (1) buat keduanya},\: \: \color{red}f(x)>0\\ &(x+1)>0\Leftrightarrow x>-1\\ &\color{red}\textrm{Syarat (2)},\: \: \log \left ( x+1 \right )<-1\\ &\log (x+1)<\log 10^{-1}\\ &x+1<\displaystyle \frac{1}{10}\Leftrightarrow x<-\frac{9}{10}\\ &\color{red}\textrm{Syarat (3)},\: \: \log \left ( x+1 \right )> 1\\ &\log (x+1)>\log 10^{1}\\ &(x+1)>10\Leftrightarrow x>9\\ &\textrm{Jadi},\: \: \color{red}-1<x<-0,9\: \: \textrm{atau}\: \: x>9 \end{aligned} \end{array}$