Berikut ini adalah kumpulan soal mengenai persamaan logaritma Tipe SBMPTN. Jika ingin bertanya soal, silahkan gabung ke grup Facebook atau Telegram.
No. 1 Jika
x_1 dan
x_2 memenuhi
\left(^{(x-1)}\log4\right)^2=4 , maka nilai
{x_1+x_2} adalah
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 \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}
\(\begin{aligned}
x_1+x_2&=3+1\dfrac12\\
&=\color{blue}{\boxed{\boxed{\color{black}{4\dfrac12}}}}
\end{aligned}\)
No. 2 Jika x_1 dan x_2 memenuhi \left({^{x-2}\log}9\right)^2=4 , maka nilai x_1+x_2 adalah ....Ganesha Operation
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} x_2-2=9^{-\frac12} x_1+x_2=5+2\dfrac13=7\dfrac13 No. 3 Jika
x_1 dan
x_2 memenuhi
{\left({^{27}\negthinspace\log}\dfrac1{x+1}\right)^2=\dfrac19} , maka nilai
x_1x_2 adalah ....
\dfrac53 \dfrac43 \dfrac13 SBMPTN 2018 Kode 517
Alternatif Penyelesaian \(\begin{aligned}
\left({^{27}\negthinspace\log}\dfrac1{x+1}\right)^2&=\dfrac19\\[8pt]
{^{27}\negthinspace\log}\dfrac1{x+1}&=\pm\dfrac13\\[8pt]
\dfrac1{x+1}&=27^{\pm\frac13}\\
&=\left(3^3\right)^{\pm\frac13}\\
&=3^{\pm1}\\
x+1&=\dfrac1{3^{\pm1}}\\
x&=-1+\dfrac1{3^{\pm1}}
\end{aligned}\)
No. 4 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 ....
6\ ^3\negthinspace\log p 6\ ^3\negthinspace\log q
3\ ^3\negthinspace\log p
-3\ ^3\negthinspace\log q -3\ ^3\negthinspace\log q
http://www.learncy.net/problem/166/
Alternatif Penyelesaian \(\begin{aligned}
^9\negthinspace\log q^8 +{^3\negthinspace\log p^5}&=11\\
8\ ^9\negthinspace\log q+5\ ^3\negthinspace\log p&=11\\
5\ ^9\negthinspace\log q+5\ ^3\negthinspace\log p&=25\qquad-\\\hline
3\ ^9\negthinspace\log q&=-14\\
^9\negthinspace\log q&=-\dfrac{14}3
\end{aligned}\)
No. 5 Jika xy= 90 dan \log x-\log y= 1 , maka x-y= ....Alternatif Penyelesaian Syarat:
\(\begin{aligned}
\log x-\log y&= 1\\
\log\dfrac{x}y&=\log10\\
\dfrac{x}y&=10\\
x&=10y
\end{aligned}\)
\(\begin{aligned}
xy&=90\\
(10y)y&=90\\
10y^2&=90\\
y^2&=9\\
y&=\boxed{3}
\end{aligned}\)
\(\begin{aligned}
x&=10y\\
&=10(3)\\
&=\boxed{30}
\end{aligned}\)
\(\begin{aligned}
x-y&=30-3\\
&=\boxed{\boxed{27}}
\end{aligned}\)
No. 6 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= ....
Alternatif Penyelesaian \(\begin{aligned}
\log\left(x^2\right)+\log\left(10x^2\right)+\log\left(10^2x^2\right)+\cdots+\log\left(10^9x^2\right)&=55\\
\log\left(x^2\cdot10x^2\cdot10^2x^2\cdots10^9x^2\right)&=55\\
\log\left(10^{1+2+\cdots+9}x^{20}\right)&=55\\
\log\left(10^{45}x^{20}\right)&=55\\
10^{45}x^{20}&=10^{55}\\
x^{20}&=\dfrac{10^{55}}{10^{45}}\\
&=10^{10}\\
x&=10^{\frac{10}{20}}\\
&=10^{\frac12}\\
&=\boxed{\boxed{\sqrt{10}}}
\end{aligned}\)
No. 7 Penyelesaian dari
(2x)^{1+\log_22x}\geq64x^3 adalah
0\lt x\leq\dfrac14 \dfrac14\leq x\leq4 x\leq\dfrac14 atau x\geq4 0\lt x\leq\dfrac14 atau x\geq4 \dfrac14\leq x\leq2 atau x\gt4 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
\(\begin{aligned}
p^2+p&\geq3+3p\\
p^2-2p-3&\geq0\\
(p+1)(p-3)&\geq0
\end{aligned}\)
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 No. 8
Jika
x memenuhi persamaan
{{^5\negmedspace\log 5x} + {^4\negmedspace\log 4x} = {^{25}\negmedspace\log 25x^2}} maka nilai
{^x\negmedspace\log 4} =
Alternatif Penyelesaian
\(\eqalign{
{^5\negmedspace\log 5x} + {^4\negmedspace\log 4x} &= {^{25}\negmedspace\log 25x^2}\\
{^5\negmedspace\log 5x} + {^4\negmedspace\log 4}+{^4\negmedspace\log x} &= {^{5^2}\negmedspace\log (5x)^2}\\
{^5\negmedspace\log 5x} + 1+{^4\negmedspace\log x} &= {^5\negmedspace\log 5x}\\
1+{^4\negmedspace\log x} &=0\\
{^4\negmedspace\log x} &=\boxed{\boxed{-1}}
}\)
No. 9
Jika
{{^4\negmedspace\log \sqrt{x}}+ {^2\negmedspace\log y}={^4\negmedspace\log z^2}} , maka
z^2 =
x\sqrt{y} \sqrt{x}y \sqrt{x}y^2
Alternatif Penyelesaian
\(\eqalign{
{^4\negmedspace\log \sqrt{x}}+ {^2\negmedspace\log y}&={^4\negmedspace\log z^2}\\
{^4\negmedspace\log \sqrt{x}}+ {^{2^2}\negmedspace\log y^2}&={^4\negmedspace\log z^2}\\
{^4\negmedspace\log \sqrt{x}}+ {^4\negmedspace\log y^2}&={^4\negmedspace\log z^2}\\
{^4\negmedspace\log \sqrt{x}y^2}&={^4\negmedspace\log z^2}\\
\sqrt{x}y^2&=z^2\\
z^2&=\boxed{\boxed{\sqrt{x}y^2}}
}\)
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