SNBT Zone : Persamaan Logaritma
Table of Contents
Tipe:
No.
Jika x1 dan x2 memenuhi- 3
- \(3\dfrac12\)
- 4
- \(4\dfrac12\)
- 5
ALTERNATIF PENYELESAIAN
Syarat:
\(\begin{aligned} x-1&\gt0\\ x&\gt1 \end{aligned}\)
\(\begin{aligned} \left(^{(x-1)}\log4\right)^2&=4\\ ^{(x-1)}\log4&=\pm2 \end{aligned}\)
\(\begin{aligned} x_1+x_2&=3+1\dfrac12\\ &=\color{blue}{\boxed{\boxed{\color{black}{4\dfrac12}}}} \end{aligned}\)
\(\begin{aligned} x-1&\gt0\\ x&\gt1 \end{aligned}\)
\(\begin{aligned} \left(^{(x-1)}\log4\right)^2&=4\\ ^{(x-1)}\log4&=\pm2 \end{aligned}\)
\(\begin{aligned}
^{(x-1)}\log4&=2\\
(x-1)^2&=4\\
x^2-2x+1&=4\\
x^2-2x-3&=0\\
(x+1)(x-3)&=0
\end{aligned}\) | \(\begin{aligned}
^{(x-1)}\log4&=-2\\
(x-1)^{-2}&=4\\
\dfrac1{(x-1)^2}&=4\\
(x-1)^2&=\dfrac14\\
x^2-2x+1&=\dfrac14\\
4x^2-8x+4&=1\\
4x^2-8x+3&=0\\
(2x-1)(2x-3)&=0
\end{aligned}\) \(x=\dfrac12\)(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
JAWAB: D
No.
Jika x1 dan x2 memenuhiALTERNATIF PENYELESAIAN
\(\begin{aligned}
\left({^{x-2}\log}9\right)^2&=4\\
{^{x-2}\log}9&=\pm2\\
x-2&=9^{\pm\frac12}
\end{aligned}\)
- \(x_1-2=9^{\frac12}\)
\(\begin{aligned} x_1&=2+3\\ &=5 \end{aligned}\) - \(x_2-2=9^{-\frac12}\)
\(\begin{aligned} x_2&=2+\dfrac13\\ &=2\dfrac13 \end{aligned}\)
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- \(\dfrac53\)
- \(\dfrac43\)
- \(\dfrac13\)
- \(-\dfrac23\)
- \(-\dfrac43\)
ALTERNATIF PENYELESAIAN
\(\begin{aligned}
\left({^{27}\negthinspace\log}\dfrac1{x+1}\right)^2&=\dfrac19\\[3.7pt]
{^{27}\negthinspace\log}\dfrac1{x+1}&=\pm\dfrac13\\[3.7pt]
\dfrac1{x+1}&=27^{\pm\frac13}\\
&=\left(3^3\right)^{\pm\frac13}\\
&=3^{\pm1}\\
x+1&=\dfrac1{3^{\pm1}}\\
x&=-1+\dfrac1{3^{\pm1}}
\end{aligned}\)
\(\begin{aligned}
x_1&=-1+\dfrac13\\
&=-\dfrac23
\end{aligned}\)
\(\begin{aligned} x_2&=-1+\dfrac1{3^{-1}}\\ &=-1+3\\ &=2 \end{aligned}\)
\(\begin{aligned} x_1x_2&=\left(-\dfrac23\right)(2)\\ &=-\dfrac43 \end{aligned}\)
\(\begin{aligned} x_2&=-1+\dfrac1{3^{-1}}\\ &=-1+3\\ &=2 \end{aligned}\)
\(\begin{aligned} x_1x_2&=\left(-\dfrac23\right)(2)\\ &=-\dfrac43 \end{aligned}\)
Jadi, nilai x1x2 adalah \(-\dfrac43\).
JAWAB: E
JAWAB: E
No.
Jika- 6 3log p
- 6 3log q
- 3 3log p
- −3 3log p
- −3 3log q
ALTERNATIF PENYELESAIAN
\(\begin{aligned}
^9\negthinspace\log q^8 +{^3\negthinspace\log p^5}&=11\\
8\ ^9\negthinspace\log q+5\ ^3\negthinspace\log p&=11\\
5\ ^9\negthinspace\log q+5\ ^3\negthinspace\log p&=25\qquad-\\\hline
3\ ^9\negthinspace\log q&=-14\\
^9\negthinspace\log q&=-\dfrac{14}3
\end{aligned}\)
\(\begin{aligned} ^q\negthinspace\log p^2&=\dfrac{^9\negthinspace\log p^2}{^9\negthinspace\log q}\\ &=\dfrac{^{3^2}\negthinspace\log p^2}{-\dfrac{14}3}\\ &=-\dfrac3{14}{^{3}\negthinspace\log p} \end{aligned}\)
\(\begin{aligned} ^q\negthinspace\log p^2&=\dfrac{^9\negthinspace\log p^2}{^9\negthinspace\log q}\\ &=\dfrac{^{3^2}\negthinspace\log p^2}{-\dfrac{14}3}\\ &=-\dfrac3{14}{^{3}\negthinspace\log p} \end{aligned}\)
Jadi, nilai dari qlog p2 adalah \(-\dfrac3{14}{^{3}\negthinspace\log p}\).
JAWAB: -
JAWAB: -
No.
Jika- 27
- 25
- −26
- 19
- 20
ALTERNATIF PENYELESAIAN
Syarat:
\(\begin{aligned} \log x-\log y&= 1\\ \log\dfrac{x}y&=\log10\\ \dfrac{x}y&=10\\ x&=10y \end{aligned}\)
\(\begin{aligned} xy&=90\\ (10y)y&=90\\ 10y^2&=90\\ y^2&=9\\ y&=\boxed{3} \end{aligned}\)
- x > 0
- y > 0
\(\begin{aligned} \log x-\log y&= 1\\ \log\dfrac{x}y&=\log10\\ \dfrac{x}y&=10\\ x&=10y \end{aligned}\)
\(\begin{aligned} xy&=90\\ (10y)y&=90\\ 10y^2&=90\\ y^2&=9\\ y&=\boxed{3} \end{aligned}\)
\(\begin{aligned}
x&=10y\\
&=10(3)\\
&=\boxed{30}
\end{aligned}\)
\(\begin{aligned} x-y&=30-3\\ &=\boxed{\boxed{27}} \end{aligned}\)
\(\begin{aligned} x-y&=30-3\\ &=\boxed{\boxed{27}} \end{aligned}\)
Jadi, x − y = 27 .
JAWAB: A
JAWAB: A
No.
Jika- \(\dfrac1{10}\)
- \(\dfrac12\)
- 1
- \(\sqrt{10}\)
- \(2\sqrt{10}\)
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
JAWAB: D
No.
Penyelesaian dari- \(0\lt x\leq\dfrac14\)
- \(\dfrac14\leq x\leq4\)
- \(x\leq\dfrac14\) atau
x ≥ 4
- \(0\lt x\leq\dfrac14\) atau
x ≥ 4 - \(\dfrac14\leq x\leq2\) 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
\(\begin{aligned} p^2+p&\geq3+3p\\ p^2-2p-3&\geq0\\ (p+1)(p-3)&\geq0 \end{aligned}\)
Syarat:
x ≥ 4
Misal
\(\begin{aligned} p^2+p&\geq3+3p\\ p^2-2p-3&\geq0\\ (p+1)(p-3)&\geq0 \end{aligned}\)
\(\begin{aligned}
p&\leq-1\\
\log_22x&\leq-1\\
2x&\leq2^{-1}\\[3.7pt]
2x&\leq\dfrac12\\[3.7pt]
x&\leq\dfrac14
\end{aligned}\)
| 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
Jadi, penyelesaian dari (2x)1 + log2 2x ≥ 64x3 adalah \(0\lt x\leq\dfrac14\) atau x ≥ 4 .
JAWAB: D
JAWAB: D
No.
Jika x memenuhi persamaan- \(\dfrac12\)
- −2
- 2
- −1
- 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
JAWAB: D
No.
Jika \({{^4\negmedspace\log \sqrt{x}}+ {^2\negmedspace\log y}={^4\negmedspace\log z^2}}\), maka- \(x\sqrt{y}\)
- \(\sqrt{x}y\)
- \(\sqrt{x}y^2\)
- \(x^2\sqrt{y}\)
- 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
JAWAB: C
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