SNBT Zone : Persamaan Logaritma

Table of Contents
Berikut ini adalah kumpulan soal mengenai Persamaan Logaritma. Jika ingin bertanya soal, silahkan gabung ke grup Telegram, Signal, Discord, atau WhatsApp.

Tipe:

No.

Jika x1 dan x2 memenuhi ((x − 1)log 4)2 = 4, maka nilai x1 + x2 adalah ....
  1. 3
  2. \(3\dfrac12\)
  3. 4
  1. \(4\dfrac12\)
  2. 5
ALTERNATIF PENYELESAIAN
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 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

No.

Jika x1 dan x2 memenuhi ((x − 2)log 9)2 = 4, maka nilai x1 + x2 adalah ....
ALTERNATIF PENYELESAIAN
\(\begin{aligned} \left({^{x-2}\log}9\right)^2&=4\\ {^{x-2}\log}9&=\pm2\\ x-2&=9^{\pm\frac12} \end{aligned}\)

  • \(x_1-2=9^{\frac12}\)
    \(\begin{aligned} x_1&=2+3\\ &=5 \end{aligned}\)
  • \(x_2-2=9^{-\frac12}\)
    \(\begin{aligned} x_2&=2+\dfrac13\\ &=2\dfrac13 \end{aligned}\)
\(x_1+x_2=5+2\dfrac13=7\dfrac13\)
Jadi, nilai x1 + x2 adalah \(7\dfrac13\).

No.

Jika x1 dan x2 memenuhi \(\left({^{27}\negthinspace\log}\dfrac1{x+1}\right)^2=\dfrac19\), maka nilai x1x2 adalah ....
  1. \(\dfrac53\)
  2. \(\dfrac43\)
  3. \(\dfrac13\)
  1. \(-\dfrac23\)
  2. \(-\dfrac43\)
ALTERNATIF PENYELESAIAN
\(\begin{aligned} \left({^{27}\negthinspace\log}\dfrac1{x+1}\right)^2&=\dfrac19\\[3.7pt] {^{27}\negthinspace\log}\dfrac1{x+1}&=\pm\dfrac13\\[3.7pt] \dfrac1{x+1}&=27^{\pm\frac13}\\ &=\left(3^3\right)^{\pm\frac13}\\ &=3^{\pm1}\\ x+1&=\dfrac1{3^{\pm1}}\\ x&=-1+\dfrac1{3^{\pm1}} \end{aligned}\)

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

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

\(\begin{aligned} x_1x_2&=\left(-\dfrac23\right)(2)\\ &=-\dfrac43 \end{aligned}\)
Jadi, nilai x1x2 adalah \(-\dfrac43\).
JAWAB: E

No.

Jika 3log p + 9log q = 5 dan 9log q8 + 3log p5 = 11, maka nilai dari qlog p2 adalah ....
  1. 6 3log p
  2. 6 3log q
  3. 3 3log p
  1. −3 3log p
  2. −3 3log q
ALTERNATIF PENYELESAIAN
\(\begin{aligned} ^9\negthinspace\log q^8 +{^3\negthinspace\log p^5}&=11\\ 8\ ^9\negthinspace\log q+5\ ^3\negthinspace\log p&=11\\ 5\ ^9\negthinspace\log q+5\ ^3\negthinspace\log p&=25\qquad-\\\hline 3\ ^9\negthinspace\log q&=-14\\ ^9\negthinspace\log q&=-\dfrac{14}3 \end{aligned}\)

\(\begin{aligned} ^q\negthinspace\log p^2&=\dfrac{^9\negthinspace\log p^2}{^9\negthinspace\log q}\\ &=\dfrac{^{3^2}\negthinspace\log p^2}{-\dfrac{14}3}\\ &=-\dfrac3{14}{^{3}\negthinspace\log p} \end{aligned}\)
Jadi, nilai dari qlog p2 adalah \(-\dfrac3{14}{^{3}\negthinspace\log p}\).
JAWAB: -

No.

Jika xy = 90 dan log x − log y = 1, maka xy = ....
  1. 27
  2. 25
  3. −26
  1. 19
  2. 20
ALTERNATIF PENYELESAIAN
Syarat:
  • x > 0
  • y > 0

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

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

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

\(\begin{aligned} x-y&=30-3\\ &=\boxed{\boxed{27}} \end{aligned}\)
Jadi, xy = 27.
JAWAB: A

No.

Jika log (x2) + log (10x2) + log (102x2) + ⋯ + log (109x2) = 55, maka x = ....
  1. \(\dfrac1{10}\)
  2. \(\dfrac12\)
  3. 1
  1. \(\sqrt{10}\)
  2. \(2\sqrt{10}\)
ALTERNATIF PENYELESAIAN
\(\begin{aligned} \log\left(x^2\right)+\log\left(10x^2\right)+\log\left(10^2x^2\right)+\cdots+\log\left(10^9x^2\right)&=55\\ \log\left(x^2\cdot10x^2\cdot10^2x^2\cdots10^9x^2\right)&=55\\ \log\left(10^{1+2+\cdots+9}x^{20}\right)&=55\\ \log\left(10^{45}x^{20}\right)&=55\\ 10^{45}x^{20}&=10^{55}\\ x^{20}&=\dfrac{10^{55}}{10^{45}}\\ &=10^{10}\\ x&=10^{\frac{10}{20}}\\ &=10^{\frac12}\\ &=\boxed{\boxed{\sqrt{10}}} \end{aligned}\)
Jadi, \(x=\sqrt{10}\).
JAWAB: D

No.

Penyelesaian dari (2x)1 + log2 2x ≥ 64x3 adalah ....
  1. \(0\lt x\leq\dfrac14\)
  2. \(\dfrac14\leq x\leq4\)
  3. \(x\leq\dfrac14\) atau x ≥ 4
  1. \(0\lt x\leq\dfrac14\) atau x ≥ 4
  2. \(\dfrac14\leq x\leq2\) atau x > 4
ALTERNATIF PENYELESAIAN
\(\begin{aligned} (2x)^{1+\log_22x}&\geq64x^3\\ \log_2\left((2x)^{1+\log_22x}\right)&\geq\log_264x^3\\ \left(1+\log_22x\right)\log_22x&\geq\log_2\left(8\cdot8x^3\right)\\ \log_22x+{\log_2}^22x&\geq\log_28+\log_28x^3\\ {\log_2}^22x+\log_22x&\geq3+\log_2(2x)^3\\ {\log_2}^22x+\log_22x&\geq3+3\log_22x \end{aligned}\)
Misal \log_22x=p
\(\begin{aligned} p^2+p&\geq3+3p\\ p^2-2p-3&\geq0\\ (p+1)(p-3)&\geq0 \end{aligned}\)
\(\begin{aligned} p&\leq-1\\ \log_22x&\leq-1\\ 2x&\leq2^{-1}\\[3.7pt] 2x&\leq\dfrac12\\[3.7pt] x&\leq\dfrac14 \end{aligned}\) atau\(\begin{aligned} p&\geq3\\ \log_22x&\geq3\\ 2x&\geq2^3\\ 2x&\geq8\\ x&\geq4 \end{aligned}\)

Syarat:
  • 2x\gt0
    x\gt0
  • 2x\neq1
    x\neq\dfrac12
\(0\lt x\leq\dfrac14\) atau x ≥ 4
Jadi, penyelesaian dari (2x)1 + log2 2x ≥ 64x3 adalah \(0\lt x\leq\dfrac14\) atau x ≥ 4.
JAWAB: D

No.

Jika x memenuhi persamaan 5log 5x + 4log 4x = 25log 25x2 maka nilai xlog 4 = ....
  1. \(\dfrac12\)
  2. −2
  3. 2
  1. −1
  2. 1
ALTERNATIF PENYELESAIAN
\(\eqalign{ {^5\negmedspace\log 5x} + {^4\negmedspace\log 4x} &= {^{25}\negmedspace\log 25x^2}\\ {^5\negmedspace\log 5x} + {^4\negmedspace\log 4}+{^4\negmedspace\log x} &= {^{5^2}\negmedspace\log (5x)^2}\\ {^5\negmedspace\log 5x} + 1+{^4\negmedspace\log x} &= {^5\negmedspace\log 5x}\\ 1+{^4\negmedspace\log x} &=0\\ {^4\negmedspace\log x} &=-1\\ {^x\negmedspace\log 4} &=\dfrac1{-1}\\ &=\boxed{\boxed{-1}} }\)
Jadi, xlog 4 = −1.
JAWAB: D

No.

Jika \({{^4\negmedspace\log \sqrt{x}}+ {^2\negmedspace\log y}={^4\negmedspace\log z^2}}\), maka z2 = ....
  1. \(x\sqrt{y}\)
  2. \(\sqrt{x}y\)
  3. \(\sqrt{x}y^2\)
  1. \(x^2\sqrt{y}\)
  2. xy
ALTERNATIF PENYELESAIAN
\(\eqalign{ {^4\negmedspace\log \sqrt{x}}+ {^2\negmedspace\log y}&={^4\negmedspace\log z^2}\\ {^4\negmedspace\log \sqrt{x}}+ {^{2^2}\negmedspace\log y^2}&={^4\negmedspace\log z^2}\\ {^4\negmedspace\log \sqrt{x}}+ {^4\negmedspace\log y^2}&={^4\negmedspace\log z^2}\\ {^4\negmedspace\log \sqrt{x}y^2}&={^4\negmedspace\log z^2}\\ \sqrt{x}y^2&=z^2\\ z^2&=\boxed{\boxed{\sqrt{x}y^2}} }\)
Jadi, \(z^2=\sqrt{x}y^2\).
JAWAB: C



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