Contoh Soal Logaritma

\color{blue}\begin{array}{ll}\\ \fbox{1}.&\textrm{Nyatakanlah dalam bentuk logaritma}\\ &\begin{array}{lllllllll}\\ \textrm{a}.&\displaystyle 3^{3}=27&\textrm{b}.&\displaystyle 3^{-3}=\displaystyle \frac{1}{27}&\textrm{c}.&\displaystyle \sqrt{3}^{3}=3\sqrt{3}&\textrm{d}.&\displaystyle 0,001=10^{-3}\\ \textrm{e}.&\displaystyle 2^{x}=y&\textrm{f}.&\displaystyle 10^{2x}=t&\textrm{g}.&\displaystyle \sqrt[3]{4}=\displaystyle 2^{\frac{2}{3}}&\textrm{h}.&\displaystyle \displaystyle 27^{\frac{2}{3}}=9\\ \end{array}\\\\ &\textrm{Jawab}\\ &\begin{array}{|l|l|l|l|}\hline \textrm{a}.\: ^{3}\log 27=3&\textrm{b}.\: ^{3}\log \displaystyle \frac{1}{27}=-3&\textrm{c}.\: ^{\sqrt{3}}\log 3\sqrt{3}=3&\textrm{d}.\: ^{10}\log 0,001=-3\\\hline \textrm{e}.\: ^{2}\log y=x&\textrm{f}.\: ^{10}\log t=2x&\textrm{g}.\: ^{2}\log \sqrt[3]{4}=\displaystyle \frac{2}{3}&\textrm{h}.\: ^{27}\log 9=\displaystyle \frac{2}{3}\\\hline \end{array} \end{array}.

\color{blue}\begin{array}{ll}\\ \fbox{2}.&\textrm{Nyatakanlah bentuk logaritma berikut ke bentuk pangkat}\\ &\begin{array}{lllllllll}\\ \textrm{a}.&\displaystyle ^{3}\log 9=2&\textrm{b}.&\displaystyle ^{3}\log 9\sqrt{9}=3&\textrm{c}.&\displaystyle ^{3\sqrt{3}}\log 9=\frac{4}{3}&\textrm{d}.&^{\frac{1}{3}}\displaystyle \log \displaystyle \frac{1}{9}=2\\ \textrm{e}.&\displaystyle ^{\frac{1}{9}}\log \displaystyle \frac{1}{3}=\frac{1}{2}&\textrm{f}.&\displaystyle ^{3}\log 1=0&\textrm{g}.&\displaystyle ^{3}\log a=b^{3}&\textrm{h}.&\displaystyle \displaystyle ^{\frac{1}{8}}\log 2=-\frac{1}{3}\\ \end{array}\\ &\textrm{Jawab}:\\\\ &\begin{array}{|ll||ll||ll||ll|}\hline \textrm{a}.&3^{2}=9&\textrm{b}.&3^{3}=9\sqrt{9}&c.&\left (3\sqrt{3} \right )^{\frac{4}{3}}=9&d.&\left (\displaystyle \frac{1}{3} \right )^{2}=\displaystyle \frac{1}{9}\\\hline \textrm{e}.&\left ( \displaystyle \frac{1}{9} \right )^{\frac{1}{2}}=\displaystyle \frac{1}{3}&f.&3^{0}=1&g.&\displaystyle 3^{\left ( \displaystyle b^{3} \right )}=a&h.&\left ( \displaystyle \frac{1}{8} \right )^{-\frac{1}{3}}=2\\\hline \end{array} \end{array}.

\color{blue}\begin{array}{ll}\\ \fbox{3}.&\textrm{Tentukanlah nilai x yang memenuhi}\\ &\begin{array}{lllllllll}\\ \textrm{a}.&^{5}\log x=2&\textrm{b}.&\displaystyle ^{2}\log 32=x&\textrm{c}.&\displaystyle ^{6}\log \, (x+1)=2&d.&^{x}\displaystyle \log 8=\displaystyle \frac{1}{3}\\ \textrm{e}.&\displaystyle ^{x}\log \displaystyle \frac{1}{125}=3&\textrm{f}.&\displaystyle ^{\frac{1}{2}}\log x=2&\textrm{g}.&\displaystyle ^{x}\log 11\sqrt{11}=\displaystyle \frac{3}{2}&\textrm{h}.&\displaystyle \displaystyle ^{\frac{1}{8}}\log 2=x\\ \end{array}\\ &\begin{array}{|l|l||l|l||l|l||l|l|}\hline \textrm{a}.&x=5^{2}=25&\textrm{b}.&\begin{aligned}2^{x}&=32\\ 2^{x}&=2^{5}\\ x&=5\\ & \end{aligned}&\textrm{c}.&\begin{aligned}(x+1)&=6^{2}\\ (x+1)&=36\\ (x+1)&=35+1\\ x&=35 \end{aligned}&\textrm{d}.&\begin{aligned}x^{\frac{1}{3}}&=8\\ x&=8^{\frac{3}{1}}\\ x&=8^{3}\\ x&=512 \end{aligned}\\\hline \textrm{e}.&\begin{aligned}x^{3}&=\displaystyle \frac{1}{125}\\ x^{3}&=\displaystyle \left ( \frac{1}{5} \right )^{3}\\ x&=\displaystyle \frac{1}{5}\\ &\\ & \end{aligned}&\textrm{f}.&x=\left ( \displaystyle \frac{1}{2} \right )^{2}=\displaystyle \frac{1}{4}&\textrm{g}.&\begin{aligned}x^{\frac{3}{2}}&=11\sqrt{11}\\ x^{\frac{3}{2}}&=11^{\frac{3}{2}}\\ x&=11\\ &\\ &\\ & \end{aligned}&\textrm{h}.&\begin{aligned}\left ( \frac{1}{8} \right )^{x}&=2\\ \left ( 2^{-3} \right )^{x}&=2^{1}\\ 2^{-3x}&=2^{1}\\ -3x&=1\\ x&=-\displaystyle \frac{1}{3} \end{aligned}\\\hline \end{array} \end{array}.

\color{blue}\begin{array}{ll}\\ \fbox{4}.&\textrm{Tentukanlah nilai logaritma berikut}\\ &\begin{array}{lllllllll}\\ \textrm{a}.&^{5}\log 25\sqrt{5}&\textrm{b}.&\displaystyle ^{2}\log \displaystyle \frac{1}{32}&\textrm{c}.&\displaystyle ^{\sqrt{3}}\log 81\\ \textrm{d}.&^{\frac{1}{3}}\displaystyle \log \displaystyle \frac{1}{243}& \textrm{e}.&\displaystyle ^{\sqrt{2}}\log 16&\textrm{f}.&\displaystyle ^{\sqrt{5}}\log \sqrt{125}\\ \textrm{g}.&\displaystyle ^{\sqrt{\sqrt{2}}}\log \sqrt{8\sqrt{8}}&\textrm{h}.&\displaystyle \displaystyle ^{0,333...}\log \left (0,111... \right )\\ \end{array}\\\\ &\textrm{Jawab}:\\ &\begin{array}{|c|c|}\hline \begin{aligned}\textrm{a}.\quad ^{5}\log 25\sqrt{5}&=\: ^{5^{1}}\log 5^{2}.5^{\frac{1}{2}}\\ &=\: ^{5^{1}}\log 5^{\frac{5}{2}}\\ &=\displaystyle \frac{\frac{5}{2}}{1}\times \: ^{5}\log 5\\ &=\displaystyle \frac{5}{2} \end{aligned}&\begin{aligned}\textrm{b}.\quad ^{2}\log \displaystyle \frac{1}{32}&=\: ^{2^{1}}\log 2^{-5}\\ &=\displaystyle \frac{-5}{1}\times \: ^{2}\log 2\\ &=-5\\ &\\ & \end{aligned}\\\hline \begin{aligned}\textrm{c}.\quad ^{\sqrt{3}}\log 81&=\: ^{\displaystyle 3^{\frac{1}{2}}}\log 3^{4}\\ &=\displaystyle \frac{4}{\frac{1}{2}}\times \: ^{3}\log 3\\ &=8 \end{aligned}&\begin{aligned}\textrm{d}.\quad ^{\frac{1}{3}}\log \displaystyle \frac{1}{243}&=\: ^{\left (\frac{1}{3} \right )^{1}}\log \displaystyle \left (\frac{1}{3} \right )^{5}\\ &=\displaystyle \frac{5}{1}\times \: ^{\frac{1}{3}}\log \frac{1}{3}\\ &=5 \end{aligned}\\\hline \begin{aligned}\textrm{e}.\quad ^{\sqrt{2}}\log 16&=\: ^{\displaystyle 2^{\frac{1}{2}}}\log 2^{4}\\ &=\displaystyle \frac{4}{\frac{1}{2}}\times \: ^{2}\log 2\\ &=8\end{aligned}&\begin{aligned}\textrm{f}.\quad ^{\sqrt{5}}\log \sqrt{125}&=\: ^{\sqrt{5}^{1}}\log \left ( \sqrt{5} \right )^{3}\\ &=\displaystyle \frac{3}{1}\times \: ^{\sqrt{5}}\log \sqrt{5}\\ &=3\\ \end{aligned} \\\hline \begin{aligned}\textrm{g}.\quad ^{\sqrt{\sqrt{2}}}\log \sqrt{8\sqrt{8}}\\ &=\: ^{\sqrt[4]{2}}\log \left ( 8\left ( 8 \right )^{\frac{1}{2}} \right )^{\frac{1}{2}}\\ &=\: ^{2^{\frac{1}{4}}}\log 8^{\left (\frac{1}{2}+\frac{1}{4} \right )}\\ &=\: ^{2^{\frac{1}{4}}}\log 2^{3\left ( \frac{3}{4} \right )}\\ &=\displaystyle \frac{\frac{9}{4}}{\frac{1}{4}}\times \: ^{2}\log 2\\ &=9 \end{aligned}&\begin{aligned}\textrm{h}.\quad ^{0,333...}\log 0,111...\\ &=\: ^{\frac{1}{3}}\log \frac{1}{9}\\ &=\: ^{\frac{1}{3}}\log \left (\frac{1}{3} \right )^{2}\\ &=2\\ &\\ &\\ & \end{aligned} \\\hline \end{array} \end{array}.

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