Latihan Soal 1 Persiapan PAS Gasal Matematika Peminatan Kelas X (Fungsi Eksponen dan Fungsi Logaritma)

 $\begin{array}{l}\\ 1.& \text{Hasil dari}{\displaystyle \frac{3^4{.5}^3{.7}^2}{27.125.49}\text{adalah... .}}\\ & \begin{array}{l}\\ \text{a}.& 2\\ \text{b}.& 3\\ \text{c}.& 4\\ \text{d}.& 9\\ \text{e}.& 16\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}{\displaystyle \frac{3^4{.5}^3{.7}^2}{27.125.49}}& ={\displaystyle \frac{3^4{.5}^3{.7}^2}{3^3{.5}^3{.7}^2}}\\ & =3^{4-3}{.5}^{3-3}{.7}^{2-2}\\ & =3^1{.5}^0{.7}^0\\ & =3.1.1\\ & =3\end{array}\end{array}$

$\begin{array}{l}\\ 2.& \text{Bentuk sederhana dari}{\displaystyle \frac{5a^4{b}^2}{a^2{b}^{-3}}\text{adalah... .}}\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle a{b}^2}\\ \\ \text{b}.& {\displaystyle a^2{b}^2}\\ \\ \text{c}.& {\displaystyle 5a^2{b}^5}\\ \\ \text{d}.& 5a^4{b}^5\\ \\ \text{e}.& 5a^5{a}^2\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}{\displaystyle \frac{5a^4{b}^2}{a^2{b}^{-3}}}& =5a^{4-2}{b}^{2-(-3)}\\ & =5a^2{b}^5\end{array}\end{array}$

$\begin{array}{l}\\ 3.& \text{Bentuk sederhana dari}{\left({\displaystyle \frac{a{b}^2}{a^2{b}^3}}\right)}^4\text{adalah... .}\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle \frac{1}{b}}\\ \\ \text{b}.& {\displaystyle \frac{1}{ab}}\\ \\ \text{c}.& {\displaystyle \frac{1}{a^2{b}^2}}\\ \\ \text{d}.& {\displaystyle \frac{1}{a{b}^2}}\\ \\ \text{e}.& {\displaystyle \frac{1}{a^4{b}^4}}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}{\left({\displaystyle \frac{a{b}^2}{a^2{b}^3}}\right)}^4& ={\left(a^{1-2}{b}^{2-3}\right)}^4\\ & ={\left(a^{-1}{b}^{-1}\right)}^4\\ & ={\left({\displaystyle \frac{1}{ab}}\right)}^4\\ & ={\displaystyle \frac{1}{a^4{b}^4}}\end{array}\end{array}$

$\begin{array}{l}\\ 4.& \text{Bentuk sederhana dari}{\displaystyle \frac{3^{\frac{5}{6}}\times {12}^{\frac{7}{12}}}{2^{-\frac{1}{4}}\times 6^{\frac{2}{3}}}\text{adalah}....}\\ & \begin{array}{l}\\ \text{a}.6^{\frac{1}{4}}& & \text{d}.{\left({\displaystyle \frac{2}{3}}\right)}^{\frac{3}{4}}\\ \text{b}.6^{\frac{3}{4}}& \text{c}.6^{\frac{2}{3}}& \text{e}.{\left({\displaystyle \frac{3}{2}}\right)}^{\frac{3}{4}}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}{\displaystyle \frac{3^{\frac{5}{6}}\times {12}^{\frac{7}{12}}}{6^{\frac{2}{3}}\times 2^{-\frac{1}{4}}}}& ={\displaystyle \frac{3^{\frac{5}{6}}\times (3\times 4{)}^{\frac{7}{12}}}{(2\times 3{)}^{\frac{2}{3}}\times 2^{-\frac{1}{4}}}}\\ & ={\displaystyle \frac{3^{\frac{5}{6}}\times 3^{\frac{7}{12}}\times 4^{\frac{7}{12}}}{2^{\frac{2}{3}}\times 3^{\frac{2}{3}}\times 2^{-\frac{1}{4}}}}\\ & ={\displaystyle \frac{3^{\frac{5}{6}}\times 3^{\frac{7}{12}}\times {\left(2^2\right)}^{\frac{7}{12}}}{2^{\frac{2}{3}}\times 3^{\frac{2}{3}}\times 2^{-\frac{1}{4}}}}\\ & ={\displaystyle \frac{3^{\frac{5}{6}}\times 3^{\frac{7}{12}}\times 2^{\frac{14}{12}}}{2^{\frac{2}{3}}\times 3^{\frac{2}{3}}\times 2^{-\frac{1}{4}}}}\\ & =2^{(\frac{14}{12}-\frac{2}{3}+\frac{1}{4})}\times 3^{(\frac{5}{6}+\frac{7}{12}-\frac{2}{3})}\\ & =2^{(\frac{14-8+3}{12})}\times 3^{(\frac{10+7-8}{12})}\\ & =2^{\frac{9}{12}}\times 3^{\frac{9}{12}}\\ & =(2\times 3{)}^{\frac{9}{12}}\\ & =6^{\frac{9}{12}}\\ & =6^{\frac{3}{4}}\end{array}\end{array}$

$\begin{array}{l}\\ 5.& \text{Bentuk sederhana dari}{\left({\displaystyle \frac{a^{\frac{1}{2}}{b}^{-3}}{a^{-1}{b}^{-\frac{3}{2}}}}\right)}^{\frac{3}{2}}\text{adalah... .}\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle \frac{a}{b}}\\ \\ \text{b}.& {\displaystyle \frac{b}{a}}\\ \\ \text{c}.& {\displaystyle ab}\\ \\ \text{d}.& \sqrt[a]{b}\\ \\ \text{e}.& \sqrt[4]{{\left({\displaystyle \frac{a}{b}}\right)}^9}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}{\left({\displaystyle \frac{a^{\frac{1}{2}}{b}^{-3}}{a^{-1}{b}^{-\frac{3}{2}}}}\right)}^{\frac{3}{2}}& ={\displaystyle \frac{a^{\frac{3}{4}}{b}^{\frac{-9}{2}}}{a^{-\frac{3}{2}}{b}^{-\frac{9}{4}}}}\\ & =a^{\frac{3}{4}+\frac{3}{2}}{b}^{\frac{-9}{2}+\frac{9}{4}}\\ & =a^{\frac{3+6}{4}}{b}^{\frac{-18+9}{4}}\\ & =a^{\frac{9}{4}}{b}^{-\frac{9}{4}}\\ & ={\left({\displaystyle \frac{a}{b}}\right)}^{\frac{9}{4}}\\ & =\sqrt[4]{{\left({\displaystyle \frac{a}{b}}\right)}^9}\end{array}\end{array}$.

$\begin{array}{l}\\ 6.& \text{Nilai}x\text{yang memenuhi}\\ & {\displaystyle 5^2{\left({\left({\displaystyle \frac{1}{25}}\right)}^{2x+6}\right)}^{\frac{1}{6}}={\displaystyle \frac{1}{25}\text{adalah... .}}}\\ & \begin{array}{l}\\ \text{a}.& -5\\ \text{b}.& -4\\ \text{c}.& -3\\ \text{d}.& 1\\ \text{e}.& 3\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}{\displaystyle 5^2{\left({\left({\displaystyle \frac{1}{25}}\right)}^{2x+6}\right)}^{\frac{1}{6}}}& ={\displaystyle \frac{1}{25}}\\ 25.{\displaystyle 5^2{\left({\left({\displaystyle \frac{1}{25}}\right)}^{2x+6}\right)}^{\frac{1}{6}}}& =1\\ 5^2{.5}^2{.5}^{-\frac{4x+12}{6}}& =5^0\\ 5^{2+2-\frac{2}{3}x-2}& =5^0\\ {\displaystyle 2+2-\frac{2}{3}x-2}& =0\\ 2-{\displaystyle \frac{2}{3}x}& =0\\ -{\displaystyle \frac{2}{3}x}& =-2\\ x& =3\end{array}\end{array}$

$\begin{array}{l}\\ 7.& (\mathbf{\text{UM-UGM 03}})\text{Nilai}x\text{yang memenuhi}\\ & {\left({\displaystyle \frac{1}{25}}\right)}^{x-\frac{5}{2}}=\sqrt{{\displaystyle \frac{625}{5^{2-x}}}}\text{adalah... .}\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle \frac{3}{5}}\\ \text{b}.& {\displaystyle \frac{8}{5}}\\ \text{c}.& 2\\ \text{d}.& 3\\ \text{e}.& 5\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}{\left({\displaystyle \frac{1}{25}}\right)}^{x-\frac{5}{2}}& =\sqrt{{\displaystyle \frac{625}{5^{2-x}}}}\\ 5^{-2x+5}& =5^{\frac{1}{2}(4-(2-x))}\\ -2x+5& ={\displaystyle \frac{1}{2}(4-2+x)}\\ -4x+10& =2+x\\ -5x& =2-10\\ x& ={\displaystyle \frac{-8}{-5}}\\ & =\frac{8}{5}\end{array}\end{array}$

$\begin{array}{l}\\ 8.& \text{Nilai}x\text{yang memenuhi}\\ & 2^{\frac{x}{3}-1}=16\text{adalah... .}\\ & \begin{array}{l}\\ \text{a}.& 5\\ \text{b}.& 10\\ \text{c}.& 15\\ \text{d}.& 20\\ \text{e}.& 25\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}{2}^{\frac{x}{3}-1}& =16\\ 2^{\frac{x}{3}-1}& =2^4\\ {\displaystyle \frac{x}{3}-1}& =4\\ {\displaystyle \frac{x}{3}}& =5\\ x& =15\end{array}\end{array}$

$\begin{array}{l}\\ 9.& (\mathbf{\text{SPMB 05}})\text{Nilai}x\text{yang memenuhi}\\ & {\displaystyle \frac{\sqrt[3]{(0,08{)}^{7-2x}}}{(0,2{)}^{-4x+5}}=1\text{adalah... .}}\\ & \begin{array}{l}\\ \text{a}.& -3\\ \text{b}.& -2\\ \text{c}.& -1\\ \text{d}.& 0\\ \text{e}.& 1\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}{\displaystyle \frac{\sqrt[3]{(0,08{)}^{7-2x}}}{(0,2{)}^{-4x+5}}}& =1\\ \sqrt[3]{(0,08{)}^{7-2x}}& =(0,2{)}^{-4x+5}\\ (0,2{)}^{\frac{3(7-2x)}{3}}& =(0,2{)}^{-4x+5}\\ {\displaystyle 7-2x}& =-4x+5\\ 4x-2x& =5-7\\ 2x& =-2\\ x& =-1\end{array}\end{array}$

$\begin{array}{l}\\ 10.& (\mathbf{\text{SPMB 05}})\text{Nilai}x\text{yang memenuhi}\\ & {\displaystyle \frac{\sqrt[3]{{\displaystyle \frac{1}{9^{2-x}}}}}{27}=3^{x+1}\text{adalah... .}}\\ & \begin{array}{l}\\ \text{a}.& -16\\ \text{b}.& -7\\ \text{c}.& 4\\ \text{d}.& 5\\ \text{e}.& 6\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}{\displaystyle \frac{\sqrt[3]{{\displaystyle \frac{1}{9^{2-x}}}}}{27}}& =3^{x+1}\\ \sqrt[3]{{\displaystyle \frac{1}{9^{2-x}}}}& =27\times 3^{x+1}\\ \sqrt[3]{3^{-2(2-x)}}& =3^3{.3}^{x+1}\\ 3^{\frac{-4+2x}{3}}& =3^{3+(x+1)}\\ {\displaystyle \frac{-4+2x}{3}}& =4+x\\ -4+2x& =12+3x\\ 2x-3x& =12+4\\ -x& =16\\ x& =-16\end{array}\end{array}$