$\begin{array}{ll}\\ 51.&\textrm{Penyelesaian dari}\quad 2^{\displaystyle 4^{\displaystyle x}}=4^{\displaystyle 2^{\displaystyle x}}\quad\textrm{adalah}\:....\\\\ &\textrm{Solusi}:\quad \\ &\textrm{Perhatikan bahwa}\\ &\begin{aligned}&2^{\displaystyle 4^{\displaystyle x}}=4^{\displaystyle 2^{\displaystyle x}}\Leftrightarrow 2^{\displaystyle 2^{\displaystyle 2x}}=2^{\displaystyle 2.2^{\displaystyle x}}\\ &\Leftrightarrow 2^{\displaystyle 2^{\displaystyle 2x}}=2^{\displaystyle 2^{\displaystyle 1+x}}\\ &\Leftrightarrow 2x=1+x\Leftrightarrow x=1 \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 52.&\textrm{Penyelesaian dari}\\ &\left(\displaystyle \frac{1}{x^{\displaystyle 3}} \right)^{\left( \displaystyle \frac{1}{x^{\displaystyle 4}} \right)}=\left(\displaystyle \frac{1}{\sqrt[\displaystyle 3x]{x}\:} \right)^{\left( \displaystyle \frac{1}{x} \right)}\\ &\textrm{untuk}\quad x>1\quad\textrm{adalah}\:....\\ &\begin{array}{llll}\\ \textrm{a}.&\displaystyle \displaystyle \frac{4}{3}\\ \textrm{b}.&\displaystyle 2\\ \textrm{c}.&\displaystyle \frac{3}{2}\\ \textrm{d}.&\displaystyle 5\\ \textrm{e}.&\displaystyle 3 \end{array}\\\\ &\textrm{Jawab}:\quad \textbf{e}\\ &\textrm{Perhatikan bahwa}\\ &\begin{aligned}&\left(\displaystyle \frac{1}{x^{\displaystyle 3}} \right)^{\left( \displaystyle \frac{1}{x^{\displaystyle 4}} \right)}=\left(\displaystyle \frac{1}{\sqrt[\displaystyle 3x]{x}\:} \right)^{\left( \displaystyle \frac{1}{x} \right)}\:\textrm{dengan}\quad x>1\\ &\Leftrightarrow \left(\displaystyle \frac{1}{x^{\displaystyle 3}} \right)^{\displaystyle x}=\left(\displaystyle \frac{1}{x^{\displaystyle \frac{1}{3x}}} \right)^{\displaystyle x^{\displaystyle 4}}\\ &\Leftrightarrow \left(\displaystyle x^{\displaystyle -3} \right)^{\displaystyle x}=\left(x^{\displaystyle -\frac{1}{3x}} \right)^{\displaystyle x^{\displaystyle 4}}\\ &\Leftrightarrow x^{\displaystyle -3x}=x^{\displaystyle -\frac{x^{\displaystyle 4}}{3x}}\\ &\Leftrightarrow -3x=\displaystyle -\frac{x^{\displaystyle 4}}{3x}\Leftrightarrow 9x^{\displaystyle 2}=x^{\displaystyle 4}\\ &\Leftrightarrow x^{\displaystyle 2}=9\Leftrightarrow x=\left| 3 \right|=\pm 3,\quad\textrm{karena}\quad x>1\\ &\textrm{maka nilai}\quad x=3 \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 53.&\textrm{Penyelesaian dari}\quad x^{\displaystyle x^{\displaystyle 4}}=\displaystyle 4\quad\textrm{adalah}\:....\\\\ &\textrm{Solusi}:\quad \\ &\textrm{Perhatikan bahwa}\\ &\begin{aligned}&x^{\displaystyle x^{\displaystyle 4}}=\displaystyle 4\\ &\textrm{masing-masing ruas dipangkatkan}\quad 4,\\ &\textrm{selanjutnya}\\ &\left( x^{\displaystyle x^{\displaystyle 4}} \right)^{\displaystyle 4}=4^{\displaystyle 4}\Leftrightarrow \left( x^{\displaystyle 4} \right)^{\left( \displaystyle x^{\displaystyle 4} \right)}=4^{\displaystyle 4}\\ &\Leftrightarrow x^{\displaystyle 4}=4\Leftrightarrow \left| x \right|=\sqrt[\displaystyle 4]{4}=\sqrt[\displaystyle 4]{2^{\displaystyle 2}}\\ &\Leftrightarrow \left| x \right|=\sqrt{2}\\ &\textrm{Jadi, nilai}\quad x=-\sqrt{2}\quad \textrm{atau}\quad x=\sqrt{2} \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 54.&\textrm{Jika diketahui}\quad x^{\displaystyle 2x^{\displaystyle 6}}=\displaystyle 3\quad\textrm{dengan}\quad x>1,\\ &\textrm{maka nilai}\quad \left( x^{\displaystyle x^{\displaystyle x^{\displaystyle 6}}} \right)^{\displaystyle \sqrt{3}}=\:....\\\\ &\textrm{Solusi}:\quad \\ &\textrm{Perhatikan bahwa}\\ &\begin{aligned}&x^{\displaystyle 2x^{\displaystyle 6}}=\displaystyle 3\\ &\textrm{masing-masing ruas dipangkatkan}\quad 3,\\ &\textrm{selanjutnya}\\ &x^{\displaystyle 3.2x^{\displaystyle 6}}=\displaystyle 3^{\displaystyle 3}\Leftrightarrow \left( x^{\displaystyle 6} \right)^{\left( \displaystyle x^{\displaystyle 6} \right)}=3^{\displaystyle 3}\\ &\Leftrightarrow x^{\displaystyle 6}=3\Leftrightarrow x^{\displaystyle 2.3}=3\\ &\Leftrightarrow x^{\displaystyle 3}=3^{\displaystyle \frac{1}{2}}\Leftrightarrow x^{\displaystyle 3}=\sqrt{3}.\quad\textrm{dengan}\quad x>1\\ &\textrm{Sehingga}\\ &\left( x^{\displaystyle x^{\displaystyle x^{\displaystyle 6}}} \right)^{\displaystyle \sqrt{3}}=\left( x^{\displaystyle x^{\displaystyle 3}} \right)^{\sqrt{3}}=\left( x^{\displaystyle \sqrt{3}} \right)^{\displaystyle \sqrt{3}}\\ &=x^{\displaystyle 3}=\sqrt{3}\\ &\textrm{Jadi, nilai}\quad \left( x^{\displaystyle x^{\displaystyle x^{\displaystyle 6}}} \right)^{\displaystyle \sqrt{3}}=\sqrt{3} \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 55.&\textrm{Penyelesaian dari}\quad x^{\displaystyle x}=3^{\displaystyle x+3^{\displaystyle 2}}\quad\textrm{adalah}\:....\\ &\begin{array}{llll}\\ \textrm{a}.&\displaystyle 3\\ \textrm{b}.&\displaystyle 3^{\displaystyle \sqrt{3}}\\ \textrm{c}.&\displaystyle 6\\ \textrm{d}.&\displaystyle 9\\ \textrm{e}.&\displaystyle 3^{\displaystyle 3} \end{array}\\\\ &\textrm{Jawab}:\quad \textbf{d}\\ &\textrm{Perhatikan bahwa}\\ &\begin{aligned}&x^{\displaystyle x}=3^{\displaystyle x+3^{\displaystyle 2}}\Leftrightarrow x^{\displaystyle x}=3^{\displaystyle x+9}\\ &\Leftrightarrow x^{\displaystyle x}=3^{\displaystyle x}.3^{\displaystyle 9}\\ &\Leftrightarrow \displaystyle \frac{x^{\displaystyle x}}{3^{\displaystyle x}}=3^{\displaystyle 9}\Leftrightarrow \left( \displaystyle \frac{x}{3} \right)^{\displaystyle x}=3^{\displaystyle 9}\\ &\textrm{masing-masing ruas dipangkatkan}\quad \displaystyle \frac{1}{3},\\ &\textrm{selanjutnya}\\ &\Leftrightarrow \left( \left( \displaystyle \frac{x}{3} \right)^{\displaystyle x} \right)^{\displaystyle \frac{1}{3}}=\left( 3^{\displaystyle 9} \right)^{\displaystyle \frac{1}{3}}\\ &\Leftrightarrow \left( \displaystyle \frac{x}{3} \right)^{\left( \displaystyle \frac{x}{3} \right)}=3^{\displaystyle 3}\\ &\Leftrightarrow \displaystyle \frac{x}{3}=3\Leftrightarrow x=9\\ &\textrm{Jadi, nilai}\quad x=9 \end{aligned} \end{array}$.
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