$\begin{array}{ll}\\ 102.&\textrm{Agar}\: \: \log \left ( x^{2}-1 \right )<0\: \: \textrm{maka}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&-1<x<1\\ \textrm{b}.&-\sqrt{2}<x<\sqrt{2}\\ \textrm{c}.&x<-1\: \: \textrm{atau}\: \: x>1\\ \textrm{d}.&x<-\sqrt{2}\: \: \textrm{atau}\: \: x>\sqrt{2}\\ \color{red}\textrm{e}.&-\sqrt{2}<x<-1\: \: \textrm{atau}\: \: 1<x<\sqrt{2} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{e}\\ &\begin{aligned}&\log \left ( x^{2}-1 \right )<0\\ &\textrm{Diketahui}\: \: \color{red}\log f(x)<0,\: \: \color{blue}\textrm{maka}\\ &\color{purple}\textrm{Syarat (1)},\: \: \color{red}f(x)>0\\ &\Leftrightarrow x^{2}-1>0\\ &\Leftrightarrow x<-1\: \: \textrm{atau}\: \: x>1\\ &\color{purple}\textrm{Syarat (2)},\: \: \log \left ( x^{2}-1 \right )<0\\ &\log \left ( x^{2}-1 \right )<\log 1\\ &\Leftrightarrow x^{2}-1<1\\ &\Leftrightarrow x^{2}-2<0\\ &\Leftrightarrow x^{2}-\left ( \sqrt{2} \right )^{2}<0\\ &\Leftrightarrow -\sqrt{2}<x<\sqrt{2}\\ &\textrm{Jadi},\: \: \color{red}-\sqrt{2}<x<-1\: \: \color{black}\textrm{atau}\: \: \color{red}1<x<\sqrt{2} \end{aligned} \end{array}$
$\begin{array}{ll}\\ 103.&\textrm{Himpunan penyelesaian dari}\\ &^{.^{\frac{1}{2}}}\log \left ( x^{2}-3 \right )>0 \: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&\left \{ x|-2<x<-\sqrt{3}\: \: \textrm{atau}\: \: \sqrt{3}<x<2 \right \}\\ \textrm{b}.&\left \{ x|-\sqrt{3}<x<-1\: \: \textrm{atau}\: \: \sqrt{3}<x<2 \right \}\\ \textrm{c}.&\left \{ x|-2<x<-\sqrt{3} \right \}\\ \textrm{d}.&\left \{ x|-2<x<-\sqrt{3} \right \}\\ \textrm{e}.&\left \{ x|\sqrt{3}<x<2 \right \} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\begin{aligned}&^{.^{\frac{1}{2}}}\log \left ( x^{2}-3 \right )>0\\ &\textrm{Diketahui}\: \: \color{red}^{.^{\frac{1}{2}}}\log f(x)>0,\: \: \color{blue}\textrm{maka}\\ &\color{purple}\textrm{Syarat (1)},\: \: \color{red}f(x)>0\\ &\Leftrightarrow x^{2}-3>0\\ &\Leftrightarrow x<-\sqrt{3}\: \: \textrm{atau}\: \: x>\sqrt{3}\\ &\color{purple}\textrm{Syarat (2)},\: \: ^{.^{\frac{1}{2}}}\log \left ( x^{2}-3 \right )>0\\ &^{.^{\frac{1}{2}}}\log \left ( x^{2}-3 \right )>\: ^{.^{\frac{1}{2}}}\log 1\\ &\Leftrightarrow x^{2}-3<1\quad \left (\color{black}\textrm{karena basisnya}\: \: \displaystyle \frac{1}{2}<1 \right )\\ &\Leftrightarrow x^{2}-4<0\\ &\Leftrightarrow x^{2}-2^{2}>0\\ &\Leftrightarrow -2<x<2\\ &\textrm{Jadi},\: \: \color{red}-2<x<-\sqrt{3}\: \: \color{black}\textrm{atau}\: \: \color{red}\sqrt{3}<x<2 \end{aligned} \end{array}$
$\begin{array}{ll}\\ 104.&\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi}\\ &^{2}\log \left ( x^{2}-x \right )\leq 1 \: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&x<0\: \: \textrm{atau}\: \: x>1\\ \textrm{b}.&-1\leq x\leq 2,\: x\neq 1\: \: \textrm{atau}\: \: x\neq 0\\ \color{red}\textrm{c}.&-1\leq x< 0\: \: \textrm{atau}\: \: 1<x\leq 2\\ \textrm{d}.&-1< x\leq 0\: \: \textrm{atau}\: \: 1\leq x< 2\\ \textrm{e}.&-1\leq x\leq 0\: \: \textrm{atau}\: \: 1\leq x\leq 2 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\begin{aligned}&^{2}\log \left ( x^{2}-x \right )\leq 1\\ &\textrm{Diketahui}\: \: \color{red}^{2}\log f(x)>0,\: \: \color{blue}\textrm{maka}\\ &\color{purple}\textrm{Syarat (1)},\: \: \color{red}f(x)>0\\ &\Leftrightarrow x^{2}-x>0\Leftrightarrow x(x-1)>0\\ &\Leftrightarrow x<0\: \: \textrm{atau}\: \: x>1\\ &\color{purple}\textrm{Syarat (2)},\: \: ^{2}\log \left ( x^{2}-x \right )\leq 1\\ &^{2}\log \left ( x^{2}-x \right )\leq \: ^{2}\log 2\\ &\Leftrightarrow x^{2}-x\leq 2\\ &\Leftrightarrow x^{2}-x-2\leq 0\\ &\Leftrightarrow (x+1)(x-2)\leq 0\\ &\Leftrightarrow -1\leq x\leq 2\\ &\textrm{Jadi},\: \: \color{red}-1\leq x< 0\: \: \color{black}\textrm{atau}\: \: \color{red}1<x\leq 2 \end{aligned} \end{array}$
$\begin{array}{ll}\\ 105.&\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi}\\ &\left | \log (x+1) \right |> 1 \: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&x<-0,9\: \: \textrm{atau}\: \: x>9\\ \textrm{b}.&x<-9\: \: \textrm{atau}\: \: x>9\\ \color{red}\textrm{c}.&-1<x<-0,9\: \: \textrm{atau}\: \: x>9\\ \textrm{d}.&-9< x<0,9\\ \textrm{e}.&-0,9<x<9 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\begin{aligned}&\textrm{Ingat bahwa}\\ &\left | x \right |>A\Leftrightarrow \color{black}x<-A\: \: \textrm{atau}\: \: x>A,\: \: \color{red}A>0\\ &\Leftrightarrow \log (x+1)<-1\: \: \textrm{atau}\: \: \log (x+1)>1\\ &\color{red}\textrm{Syarat (1) buat keduanya},\: \: \color{red}f(x)>0\\ &(x+1)>0\Leftrightarrow x>-1\\ &\color{red}\textrm{Syarat (2)},\: \: \log \left ( x+1 \right )<-1\\ &\log (x+1)<\log 10^{-1}\\ &x+1<\displaystyle \frac{1}{10}\Leftrightarrow x<-\frac{9}{10}\\ &\color{red}\textrm{Syarat (3)},\: \: \log \left ( x+1 \right )> 1\\ &\log (x+1)>\log 10^{1}\\ &(x+1)>10\Leftrightarrow x>9\\ &\textrm{Jadi},\: \: \color{red}-1<x<-0,9\: \: \color{black}\textrm{atau}\: \: \color{red}x>9 \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 106.&\textrm{Himpunan penyelesaian pertidaksamaan}\\ &\log 4+\log (x+3)\leq \log x^{2}\\ &\textrm{a}.\quad \left \{ x|x\geq 6,\, x\in \mathbb{R} \right \}\\ &\textrm{b}.\quad \left \{ x|-3<x\leq -2\: \: \textrm{atau}\: \: x\geq 6\, x\in \mathbb{R} \right \}\\ &\textrm{c}.\quad \left \{ x|-3<x\leq -2\: \: \textrm{atau}\: \: 0\leq x\leq 6\, x\in \mathbb{R} \right \}\\ &\textrm{d}.\quad \left \{ x|x\leq -2\: \: \textrm{atau}\: \: x\geq 6,\, x\in \mathbb{R} \right \}\\ &\textrm{e}.\quad \left \{ x|x\leq -4\: \: \textrm{atau}\: \: x\geq 4 ,\, x\in \mathbb{R} \right \}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\begin{aligned}&\textrm{Diketahui}\\ &\log 4+\log (x+3)\leq \log x^{2}\\ &\textrm{adalah bentuk}\: \: \color{red}^{a}\log f(x)\leq \, ^{a}\log g(x)\\ &\color{blue}\textrm{Syarat penyelesaian ada 2}\\ &\bullet \quad \textrm{basis}:\: \: a=10>0,\, \neq 1\\ &\bullet \quad \textrm{numerus}:\begin{cases} (1) & x+3>0\Rightarrow x>-3 \\ (2) & x^{2}>0\Rightarrow x\neq 0 \end{cases}\\ &\color{blue}\textrm{Proses penyelesaian}\\ &\log 4(x+3)\leq \log x^{2}\\ &4(x+3)\leq x^{2}\Leftrightarrow x^{2}\geq 4x+12\\ &\Leftrightarrow x^{2}-4x-12\geq 0\\ &\Leftrightarrow (x+2)(x-6)\geq 0\\ &x\leq -2\: \: \textrm{atau}\: \: x\geq 6\\ &\textrm{Jadi},\, \textrm{HP}=\color{red}\left \{ x|-3<x\leq -2\: \: \textrm{atau}\: \: x\geq 6\,\: x\in \mathbb{R} \right \} \end{aligned} \end{array}$.
$.\qquad\: \begin{aligned}&\textrm{Proses penyelesaian}\\ &^{6}\log \left (x^{2}-x-6 \right )>1\\ &\Leftrightarrow \, ^{6}\log \left (x^{2}-x-6 \right )>1.\, ^{6}\log 6\\ &\Leftrightarrow \, ^{6}\log \left (x^{2}-x-6 \right )>\, ^{6}\log 6\\ &\Leftrightarrow \, ^{a}\log f(x)>\, ^{a}\log p,\\ &\textrm{Karena}\: \color{blue}\textrm{basis}=a=6, \, \color{black}\textrm{maka tanda}\\ &\textrm{pertidaksamaan tetap. Selanjutnya}\\ &f(x)>p)\\ &\Leftrightarrow x^{2}-x-6>6\\ &\Leftrightarrow x^{2}-x-12>0\\ &\Leftrightarrow (x+3)(x-4)<0\\ &\Leftrightarrow \color{red}x<-3\: \: \color{black}\textrm{atau}\: \: \color{red}x>4\\ &\textrm{Karena}\: \: -2<x\: \: \textrm{atau}\: \: x>3,\: \textrm{maka}\\ &\textrm{HP}=\left \{ \color{red}x<-3\: \: \color{black}\textrm{atau}\: \: \color{red}x>4 \right \} \end{aligned}$.
$\begin{array}{ll} 108.&\textrm{Himpunan penyelesaian dari}\\ &\textrm{pertidaksamaan bentuk}\\ &\log x^{2}< \log (x+3)+2\log 2\\ &\textrm{adalah}\: ....\\ &\begin{array}{lll} \textrm{a}.&\left \{ -3<x<0\: \: \textrm{atau}\: \: 0<x<6 \right \}\\ \textrm{b}.&\color{red}\left \{ -2<x<0\: \: \textrm{atau}\: \: 0<x<6 \right \}\\ \textrm{c}.&\left \{ -1<x<0\: \: \textrm{atau}\: \: 0<x<6 \right \}\\ \textrm{d}.&\left \{ -2<x<0\: \: \textrm{atau}\: \: 0<x<7 \right \}\\ \textrm{e}.&\left \{ -1<x<0\: \: \textrm{atau}\: \: 0<x<8 \right \}\\ \end{array}\\\\ &\textbf{Jawab}:\quad \color{red}\textbf{b}\\ &\begin{array}{|c|c|}\hline \textrm{Syarat Numerus}&\textrm{Syarat Numerus}\\\hline \begin{aligned}f(x)>0&\\ x^{2}>0&\\ x\neq 0& \end{aligned}&\begin{aligned}g(x)&>0\\ x+3&>0\\ x&>-3 \end{aligned}\\\hline \textrm{Kita pilih}&\begin{cases} & -3<x<0 \\ & \textrm{atau}\\ & x>0 \end{cases}\\\hline \end{array}\\ &\begin{aligned}&\log x^{2}< \log (x+3)+2\log 2\\ &\Leftrightarrow \log x^{2}< \log (x+3)+\log 2^{2}\\ &\Leftrightarrow \log x^{2}< \log (x+3). 2^{2}\\ &\Leftrightarrow \, ^{a}\log f(x)<\, ^{a}\log g(x),\\ &\textrm{Karena}\: \color{blue}\textrm{basis}=a=10, \, \color{black}\textrm{maka tanda}\\ &\textrm{pertidaksamaan tetap. Selanjutnya}\\ &f(x)<g(x)\\ &\Leftrightarrow x^{2}<(x+3).2^{2}\\ &\Leftrightarrow x^{2}<(x+3).4\\ &\Leftrightarrow x^{2}<4x+12\\ &\Leftrightarrow x^{2}-4x-12<0\\ &\Leftrightarrow (x+2)(x-6)<0\\ &\Leftrightarrow \color{red}-2<x<6\\ &\textrm{Karena}\: \: -3<x<0\: \: \textrm{atau}\: \: x>0,\: \textrm{maka}\\ &\textrm{HP}=\left \{ -2<x<0\: \: \textrm{atau}\: \: 0<x<6 \right \} \end{aligned} \end{array}$.
$\begin{array}{ll} 109.&\textrm{Suatu larutan memiliki konsentrasi}\\ &\textrm{ion}\: \: \textrm{H}^{+}\: \: \textrm{sebesar}\: \: 2\times 10^{-6}.\\ &\textrm{PH dari larutan tersebut adalah}\: ....\\ &(\log 2=0,3010)\\ &\begin{array}{lllllll} \textrm{a}.&\color{red}4.3&&&\textrm{d}.&5,7\\ \textrm{b}.&4,7\quad &\textrm{c}.&5,3\quad&\textrm{e}.&6,3 \end{array}\\\\ &\textbf{Jawab}:\\ &\begin{aligned} \textrm{pH}&=-\log \left [\textrm{ H}^{+} \right ]\\ &=-\log \left ( 2\times 10^{-6} \right )\\ &=-\log 2-\log 10^{-6}\\ &=-0,3010-(-6)\log 10\\ &=-0,3010+6.1\\ &=-0,3010+6\\ &=6-0,3010\\ &=5,699\quad \textrm{dibulatkan}\\ &=\color{red}5,7 \end{aligned} \end{array}$
