SBMPTN Zone : Persamaan Logaritma

Berikut ini adalah kumpulan soal mengenai Persamaan Logaritma tingkat SBMPTN. Jika ada jawaban yang salah mohon dikoreksi melalui komentar. Terima kasih.

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No. 1

Jika {{^4\negmedspace\log\sqrt{x}}+{^2\negmedspace\log y}={^4\negmedspace\log z^2}}, maka z^2 =

1. x\sqrt{y}
2. \sqrt{x}y
3. \sqrt{x}y^2

4. x^2\sqrt{y}
5. xy

Ganesha Operation

\(\eqalign{ {^4\negmedspace\log\sqrt{x}}+{^2\negmedspace\log y}&={^4\negmedspace\log z^2}\\ {^4\negmedspace\log\sqrt{x}}+{^{2^2}\negmedspace\log y^2}&={^4\negmedspace\log z^2}\\ {^4\negmedspace\log\sqrt{x}}+{^4\negmedspace\log y^2}&={^4\negmedspace\log z^2}\\ {^4\negmedspace\log\sqrt{x}y^2}&={^4\negmedspace\log z^2}\\ \sqrt{x}y^2&=z^2\\ z^2&=\boxed{\boxed{\sqrt{x}y^2}} }\)

Jadi, z^2 =\sqrt{x}y^2.
JAWAB: C

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No. 2

Jika x_1 dan x_2 memenuhi \left(^{(x-1)}\log4\right)^2=4, maka nilai {x_1+x_2} adalah

1. 3
2. 3\dfrac12
3. 4

4. 4\dfrac12
5. 5

Syarat:
\begin{aligned}
x-1&\gt0\\
x&\gt1
\end{aligned}

\begin{aligned}
\left(^{(x-1)}\log4\right)^2&=4\\
^{(x-1)}\log4&=\pm2
\end{aligned}

\begin{aligned} ^{(x-1)}\log4&=2\\ (x-1)^2&=4\\ x^2-2x+1&=4\\ x^2-2x-3&=0\\ (x+1)(x-3)&=0 \end{aligned} x=-1 (PM) atau \boxed{x=3}

\begin{aligned} ^{(x-1)}\log4&=-2\\ (x-1)^{-2}&=4\\ \dfrac1{(x-1)^2}&=4\\ (x-1)^2&=\dfrac14\\ x^2-2x+1&=\dfrac14\\ 4x^2-8x+4&=1\\ 4x^2-8x+3&=0\\ (2x-1)(2x-3)&=0 \end{aligned} x=\dfrac12 (PM) atau \boxed{x=\dfrac32=1\dfrac12}

\begin{aligned}
x_1+x_2&=3+1\dfrac12\\
&=\color{blue}{\boxed{\boxed{\color{black}{4\dfrac12}}}}
\end{aligned}

Jadi, {x_1+x_2=4\dfrac12}.
JAWAB: D

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No. 3

Jika x_1 dan x_2 memenuhi \left({^{x-2}\log}9\right)^2=4, maka nilai x_1+x_2 adalah ....

Ganesha Operation

\begin{aligned}
\left({^{x-2}\log}9\right)^2&=4\\
{^{x-2}\log}9&=\pm2\\
x-2&=9^{\pm\frac12}
\end{aligned}

• x_1-2=9^{\frac12}
  \begin{aligned}
  x_1&=2+3\\
  &=5
  \end{aligned}
• x_2-2=9^{-\frac12}
  \begin{aligned}
  x_2&=2+\dfrac13\\
  &=2\dfrac13
  \end{aligned}
  

x_1+x_2=5+2\dfrac13=7\dfrac13

Jadi, nilai x_1+x_2 adalah 7\dfrac13.

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No. 4

Jika x_1 dan x_2 memenuhi {\left({^{27}\negthinspace\log}\dfrac1{x+1}\right)^2=\dfrac19}, maka nilai x_1x_2 adalah ....

1. \dfrac53
2. \dfrac43
3. \dfrac13

4. -\dfrac23
5. -\dfrac43

SBMPTN 2018 Kode 517

\begin{aligned}
\left({^{27}\negthinspace\log}\dfrac1{x+1}\right)^2&=\dfrac19\\[8pt]
{^{27}\negthinspace\log}\dfrac1{x+1}&=\pm\dfrac13\\[8pt]
\dfrac1{x+1}&=27^{\pm\frac13}\\
&=\left(3^3\right)^{\pm\frac13}\\
&=3^{\pm1}\\
x+1&=\dfrac1{3^{\pm1}}\\
x&=-1+\dfrac1{3^{\pm1}}
\end{aligned}

\begin{aligned}
x_1&=-1+\dfrac13\\
&=-\dfrac23
\end{aligned}

\begin{aligned}
x_2&=-1+\dfrac1{3^{-1}}\\
&=-1+3\\
&=2
\end{aligned}

\begin{aligned}
x_1x_2&=\left(-\dfrac23\right)(2)\\
&=-\dfrac43
\end{aligned}

Jadi, nilai x_1x_2 adalah \dfrac43.
JAWAB: E

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No. 5

Jika ^3\negthinspace\log p+{^9\negthinspace\log q} = 5 dan ^9\negthinspace\log q^8 +{^3\negthinspace\log p^5} = 11, maka nilai dari ^q\negthinspace\log p^2 adalah ....

1. 6\ ^3\negthinspace\log p
2. 6\ ^3\negthinspace\log q
   
3. 3\ ^3\negthinspace\log p
   

4. -3\ ^3\negthinspace\log q
5. -3\ ^3\negthinspace\log q
   

http://www.learncy.net/problem/166/

\begin{aligned}
^9\negthinspace\log q^8 +{^3\negthinspace\log p^5}&=11\\
8\ ^9\negthinspace\log q+5\ ^3\negthinspace\log p&=11\\
5\ ^9\negthinspace\log q+5\ ^3\negthinspace\log p&=25\qquad-\\\hline
3\ ^9\negthinspace\log q&=-14\\
^9\negthinspace\log q&=-\dfrac{14}3
\end{aligned}

\begin{aligned}
^q\negthinspace\log p^2&=\dfrac{^9\negthinspace\log p^2}{^9\negthinspace\log q}\\
&=\dfrac{^{3^2}\negthinspace\log p^2}{-\dfrac{14}3}\\
\end{aligned}

Jadi, ^q\negthinspace\log p^2=.

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No. 6

Jika xy= 90 dan \log x-\log y= 1, maka x-y= ....

1. 27
2. 25
   
3. -26
   

4. 19
5. 20
   

Syarat:

• x\gt0
• y\gt0

\begin{aligned}
\log x-\log y&= 1\\
\log\dfrac{x}y&=\log10\\
\dfrac{x}y&=10\\
x&=10y
\end{aligned}

\begin{aligned}
xy&=90\\
(10y)y&=90\\
10y^2&=90\\
y^2&=9\\
y&=\boxed{3}
\end{aligned}

\begin{aligned}
x&=10y\\
&=10(3)\\
&=\boxed{30}
\end{aligned}

\begin{aligned}
x-y&=30-3\\
&=\boxed{\boxed{27}}
\end{aligned}

Jadi, x-y=27.
JAWAB: A

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No. 7

Jika \log\left(x^2\right)+\log\left(10x^2\right)+\log\left(10^2x^2\right)+\cdots+\log\left(10^9x^2\right)=55, maka x= ....

\begin{aligned}
\log\left(x^2\right)+\log\left(10x^2\right)+\log\left(10^2x^2\right)+\cdots+\log\left(10^9x^2\right)&=55\\
\log\left(x^2\cdot10x^2\cdot10^2x^2\cdots10^9x^2\right)&=55\\
\log\left(10^{1+2+\cdots+9}x^{20}\right)&=55\\
\log\left(10^{45}x^{20}\right)&=55\\
10^{45}x^{20}&=10^{55}\\
x^{20}&=\dfrac{10^{55}}{10^{45}}\\
&=10^{10}\\
x&=10^{\frac{10}{20}}\\
&=10^{\frac12}\\
&=\boxed{\boxed{\sqrt{10}}}
\end{aligned}

Jadi, x=\sqrt{10}.
JAWAB: D

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No. 8

Penyelesaian dari (2x)^{1+\log_22x}\geq64x^3 adalah

1. 0\lt x\leq\dfrac14
2. \dfrac14\leq x\leq4
3. x\leq\dfrac14 atau x\geq4

4. 0\lt x\leq\dfrac14 atau x\geq4
5. \dfrac14\leq x\leq2 atau x\gt4

\begin{aligned}
(2x)^{1+\log_22x}&\geq64x^3\\
\log_2\left((2x)^{1+\log_22x}\right)&\geq\log_264x^3\\
\left(1+\log_22x\right)\log_22x&\geq\log_2\left(8\cdot8x^3\right)\\
\log_22x+{\log_2}^22x&\geq\log_28+\log_28x^3\\
{\log_2}^22x+\log_22x&\geq3+\log_2(2x)^3\\
{\log_2}^22x+\log_22x&\geq3+3\log_22x
\end{aligned}
Misal \log_22x=p
\begin{aligned}
p^2+p&\geq3+3p\\
p^2-2p-3&\geq0\\
(p+1)(p-3)&\geq0
\end{aligned}

p\leq-1	atau	p\geq3
\log_22x\leq-1	atau	\log_22x\geq3
2x\leq2^{-1}	atau	2x\geq2^3
2x\leq\dfrac12	atau	2x\geq8
x\leq\dfrac14	atau	x\geq4

Syarat:

• 2x\gt0
  x\gt0
• 2x\neq1
  x\neq\dfrac12

0\lt x\leq\dfrac14 atau x\geq4

Jadi, 0\lt x\leq\dfrac14 atau x\geq4.
JAWAB: D

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