SOAL SBMPTN LOGARITMA DAN PEMBAHASAN - Kikiholmes

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1. 
   
   Mat-Das SNMPTN 2008

   1. 0
   2. 1
   3. 2
   4. 3
   5. 4
Jawaban C





2. 

   1. $\frac{1}{3}$
   2. $\frac{3}{4}$
   3. $\frac{4}{3}$
   4. 2
   5. 3
Jawaban B





3. Nilai semua $x$ yang memenuhi $^a\log^2 x \geq 8+2 ^a\log x$
   dengan bilangan $a>1$ adalah...
   
   UM UGM 2008

   1. $a^2 \leq x\leq a^4$
   2. $x\leq a^2 \text{ atau } x\geq a^4$
   3. $x\leq \frac{1}{4} \text{ atau } x\geq a^4$
   4. $x\leq \frac{1}{a^2} \text{ atau } x\geq a^4$
   5. $x\leq -2 \text{ atau } x\geq 4$
Jawaban

\begin{align*} ^a\log^2 x\geq 8+2\ ^a\log x\\ ^a\log^2 x-2\cdot ^a\log x-8\geq 0\\ \left(^a\log x-4 \right )\left(^a\log x+2 \right )\geq 0\\ \text{maka}\\ ^a\log x\geq 4 \text{ atau } ^a\log x\leq -2\\ \text{daerah penyelesaiannya adalah } x\leq -2 \text{ atau } x\geq 4\\ \\ ^a\log x \leq -2 = \frac{1}{a^2}\text{ atau }^a\log x\geq 4 \\ ^a\log x \leq -2 \rightarrow x\leq a^{-2} = \frac{1}{a^2}\text{ atau }^a\log x\geq 4 \rightarrow x\geq a^4\\ \end{align*}



4. Jika $^{ab}\log a=4$, maka $^{ab}\log \frac{\sqrt[3]{a}}{\sqrt b}=\cdots$
   
   Mat-Das SIMAK UI 2014

   1. -3
   2. $-\frac{3}{4}$
   3. $-\frac{1}{6}$
   4. $\frac{29}{42}$
   5. $\frac{17}{6}$
Jawaban E

\begin{align*} \text{Dimulai dari yang diketahui }\\ ^{ab}\log a &=4\\ \frac{^a\log a}{^a\log a + ^a\log b} &=4\\ ^a\log b = -\frac{3}{4}\text{ atau } ^a\log a + \ ^a\log b &=\frac{1}{4}\end{align*}

\begin{align*} \text{maka}\\ \ ^{ab}\log \frac{\sqrt[3]{a}}{\sqrt b} &= \ ^{ab}\log a^{\frac{1}{3}} - ^{ab}\log b^{\frac{1}{2}}\\ &= \frac{1}{3}\cdot\ ^{ab}\log a-\frac{1}{2}\cdot\ \frac{^a\log b}{^a\log a+^a\log b}\\\\ \text{ingat lagi bahwa } ^{ab}\log a &=4, \ ^a\log b =-\frac{3}{4}, ^a\log a+^a\log b =\frac{1}{4}\\\\ &= \frac{1}{3}\cdot\ 4-\frac{1}{2}\cdot\ \frac{-\frac{3}{4}}{\frac{1}{4}}\\ &= \frac{4}{3}+\frac{3}{2}\\ &= \frac{17}{6} \end{align*}



5. \[\begin{align*} \text{Jika }x>0\text{ dan } y>0, \text{ maka }\\ \frac{7-7\ \log^2 xy}{1-\log x^3 y^2 + 2\ \log x\sqrt{y}}=\cdots \end{align*}\]

   1. 7 + log xy
   2. 7 log xy
   3. 7 log 10xy
   4. $\frac{1}{7}$
   5. 7
Jawaban C

\[\begin{align*} \frac{7-7\ \log^2 xy}{1-\log x^3 y^2 + 2\ \log x\sqrt{y}} &= \frac{7-7\ \log^2 xy}{1-\log x^3y^2+2\ \log\left(x\sqrt{2} \right )^2}\\ &= \frac{7\left(1-\log^2 xy \right )}{1-\log x^3y^2+\log x^2y}\\ &= \frac{7\left(1-\log xy \right )\left(1+\log xy \right )}{1+\log x^2y - \log x^3y^2}\\ &= \frac{7\left(1-\log xy \right )\left(1+\log xy \right )}{1+\log \frac{x^2y}{x^3y^2}}\\ &= \frac{7\left(1-\log xy \right )\left(1+\log xy \right )}{1+\log \frac{1}{xy}}\\ &= \frac{7\left(1-\log xy \right )\left(1+\log xy \right )}{1+\log \frac{1}{\left (xy \right )^{-1}}}\\ &= \frac{7\left(1-\log xy \right )\left(1+\log xy \right )}{1-\log xy}\\ &= 7\left(1+\log xy \right )\\ &= 7\left(\log 10 +\log xy \right )\\ &= 7\log 10xy \end{align*}\]



6. \[\begin{align*} \text{Nilai dari }\\ \left (\frac{^5\log 7 \cdot \ ^{49}\log 625+ ^8\log 512}{^6\log 216 -^6\log 36} \right )^4=\cdots \end{align*}\]

   1. -125
   2. -50
   3. 50
   4. 125
   5. 625
Jawaban E

\[\begin{align*} \left (\frac{^5\log 7 \cdot \ ^{49}\log 625+ ^8\log 512}{^6\log 216 -^6\log 36} \right )^4 &= \left (\frac{^5\log 7\cdot\ ^{7^2}\log 5^4+^8 \log 8^3}{^6\log 6^3 - ^6\log 6^2} \right )^4\\ &= \left (\frac{\frac{4}{2} ^5\log7\cdot ^7\log5+3\ ^8\log 8}{3\ ^6\log 6 -2\ ^6\log 6} \right )^4\\ &= \left (\frac{2\ ^5\log5+3\ ^8\log8}{3-2} \right )^4\\ &= \left (\frac{2+3}{1} \right )^4\\ &= 5^4\\ &= 625 \end{align*}\]



7. Jika $\alpha$ dan $\beta$ penyelesaian persamaan $^2\log\left(^2\log\left(x+7 \right )+1 \right )=\ ^2\log\left(^2\log x+^2\log\left(x-3 \right ) \right )$,$\\$ maka $\alpha + \beta=\cdots$
   
   Mat-IPA UM UGM 2010

   1. 2
   2. 3
   3. 4
   4. 5
   5. 6
Jawaban D

\[\begin{align*} ^2\log \left(\ ^2\log\left(x+7 \right )+1\right )&=\ ^2\log\left(^2\log x+\ ^2\log\left(x-3 \right)\right)\\ ^2\log\left(x+7 \right )+\ ^2\log2 &=\ ^2\log x+\ ^2\log\left(x-3 \right )\\ ^2\log\left(x+7 \right )\left(2 \right )&=\ ^2\log\left(x \right )\left(x-3 \right )\\ 2x+14&=x^2-3x\\ x^2-5x-14&=0\\ \left(x-7 \right )\left(x+2 \right )&=0\\ \alpha =7 \text{ dan }\beta &=-2\\ \\ \text{maka, }\alpha+\beta&=5 \end{align*}\]



8. Nilai $\left (^a\log\frac{1}{b^2} \right )\left (^b\log\frac{1}{c^2} \right )\left (^c\log\frac{1}{a^3} \right )=\cdots$
   
   Mat-Das SNMPTN 2010

   1. -6
   2. -8
   3. -10
   4. -12
   5. -14
Jawaban D

\[\begin{align*}&= \left (^a\log\frac{1}{b^2} \right )\left (^b\log\frac{1}{c^2} \right )\left (^c\log\frac{1}{a^3} \right )\\\\ \text{Penyelesaian }\\ &=\left(^a\log b^{-2} \right )\cdot\left(^a\log c^{-2} \right )\cdot\left(^c\log a^{-3} \right )\\ &=-2\cdot-2\cdot-3\left(^a\log b \right )\left(^b\log c \right )\left(^c\log a \right )\\ &=-12 \end{align*}\]



9. Jika $p=\left(^a\log 2 \right )\left( ^{a^2b}\log 4\right),$ maka $\frac{1}{p}=\cdots$
   
   Mat-Das SBMPTN 2014

   1. $2\ ^2\log a+\ ^2\log\sqrt{a}\ ^2\log b$
   2. $2\ ^2\log a+\frac{1}{2}\ ^2\log \left(ab \right )$
   3. $\left(^2\log a \right )^2+\ \frac{1}{2}\ ^2\log a\cdot\ ^2\log b$
   4. $\left(^2\log a \right )^2+\ \frac{1}{2}\ ^2\log \left(ab \right )$
   5. $\left(^2\log a \right )^2+\ ^2\log \sqrt{ab}$
Jawaban C

\[\begin{align*} p &=\left(^a\log2 \right )\left(^{a^2b}\log 4 \right )\\ \frac{1}{2}&=\frac{1}{\left \{ \left(^2\log2 \right )\left(^{a^2b}\log \right ) \right \}}\\ \frac{1}{p}&=\left(^2\log a \right )\left(^4\log a^2 b \right )\\ \frac{1}{p}&=\left(^2\log a \right )\left(^{2^2}\log a^2 b \right )\\ \frac{1}{p}&=\left(^2\log a \right )\left(\frac{1}{2}\left(^2\log a^2 +\ ^2\log b\right ) \right ) \end{align*}\]\[\begin{align*} \frac{1}{p}&=\left(^2\log a \right )\left(\frac{1}{2}\left(2\ ^2\log a+\ ^2\log b \right )\right)\\ \frac{1}{p}&=\left(^2\log a \right )\left(\left(^2\log a \right )+\frac{1}{2}\left ( ^2\log b \right )\right)\\ \frac{1}{p}&=\left(^2\log a \right )^2+\frac{1}{2}\left(^2\log a \right )\left(^2\log b \right ) \end{align*}\]



10. Diketahui $f(n)=\ ^3\log4\times\ ^4\log5\cdots^{n-1}\log\text{n}$ . Jika $a_1\ \text{dan}\ a_2$ adalah penyelesaian dari $f(a)+f(a^2)+\cdots+f(a^9)=f(a)\cdot f(a^5)$ maka $a_1\cdot a_2$ adalah...
    
    Mat-Das SBMPTN 2014

    1. $3^7$
    2. $3^8$
    3. $3^9$
    4. $3^{10}$
    5. $3^{11}$
Jawaban C

\[\begin{align*}\Leftrightarrow f(n)=\ ^3\log4\cdot^4\log5\cdots^{n-1}\log n=\ ^3\log n \\ \Leftrightarrow f(a)+f(a^2)+\cdots+f(a^9)=f(a)\cdot f(a^5) \\ ^3\log a+^3\log a^2+\cdots+^3\log a^9=^3\log a\cdot^3\log a^5 \\ ^3\log (a\cdot a^2 \cdots a^9)=\ ^3\log a \ \cdot 5\ \cdot\ ^3\log a \\ ^3\log a^{45} = 5 \left(\log a \right )^2 \\ 45\cdot^3\log a=5 \left(\log a \right )^2\\ 9\cdot ^3\log a=\left(^3\log a \right ) \\ \left(^3\log a \right )^2-9\cdot ^3\log a=0\\ \left(^3\log a \right )\left(^3\log a-9 \right )=0\\\end{align*}\]
sehingga,
\[\begin{align*} ^3\log a=0\rightarrow a_1=3^0=1\\ ^3\log a-9=0\\ \rightarrow ^3\log a=9\rightarrow a_2=3^9\\ \text{Jadi,}\\ a_1 \cdot a_2=3^9 \end{align*}\]

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