Exercise Zone : Logaritma

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

Tipe:

StandarSNBTHOTS

No.

Jika 5log 3 = x dan 3log 2 = y, maka 18log 15 sama dengan
  1. \dfrac{2+x}{x(1+y)}
  2. \dfrac{1+x}{x(2+y)}
  3. \dfrac{2+y}{y(1+x)}
  1. \dfrac{1+y}{y(2+x)}
  2. \dfrac{3+x}{x(1+y)}
ALTERNATIF PENYELESAIAN
\begin{aligned} ^5\negmedspace\log 3\cdot{^3\negmedspace\log 2}&=xy\\ ^5\negmedspace\log2&=xy \end{aligned}
\begin{aligned} ^{18}\negmedspace\log 15&=\dfrac{^5\negmedspace\log15}{^5\negmedspace\log18}\\[3.7pt] &=\dfrac{^5\negmedspace\log(5\cdot3)}{^5\negmedspace\log\left(3^2\cdot2\right)}\\[3.7pt] &=\dfrac{^5\negmedspace\log5+{^5\negmedspace\log3}}{^5\negmedspace\log3^2+{^5\negmedspace\log2}}\\[3.7pt] &=\dfrac{^5\negmedspace\log5+{^5\negmedspace\log3}}{2\ {^5\negmedspace\log3}+{^5\negmedspace\log2}}\\[3.7pt] &=\dfrac{1+x}{2x+xy}\\ &=\boxed{\boxed{\dfrac{1+x}{x(2+y)}}} \end{aligned}
Jadi, ^{18}\negmedspace\log 15=\dfrac{1+x}{x(2+y)}.
JAWAB: B

No.

Jika \(\dfrac{2-6\sqrt2}{\sqrt2-6}=x\), maka xlog 0,125 = ....
ALTERNATIF PENYELESAIAN
\begin{aligned} x&=\dfrac{2-6\sqrt2}{\sqrt2-6}\cdot\dfrac{\sqrt2+6}{\sqrt2+6}\\[4pt] &=\dfrac{2\sqrt2+12-12-36\sqrt2}{2-36}\\[4pt] &=\dfrac{-34\sqrt2}{-34}\\[4pt] &=\sqrt2 \end{aligned}
\begin{aligned} {^x\negmedspace\log}\ 0{,}125&={^{\sqrt2}\negmedspace\log}\ \dfrac18\\ &={^{2^{\frac12}}\negmedspace\log}\ 2^{-3}\\ &=\dfrac21\cdot(-3)\\ &=-6 \end{aligned}
Jadi, xlog 0,125 = −6.

No.

Jika a > 1, b > 1, dan c > 1, maka \left({^a\negthinspace\log}\dfrac1b\right)\left({^b\negthinspace\log}\dfrac1c\right)\left({^c\negthinspace\log}\dfrac1a\right)= ....
  1. 1 − abc
  2. abc
  3. abc
  1. 1
  2. −1
ALTERNATIF PENYELESAIAN
\begin{aligned} \left({^a\negthinspace\log}\dfrac1b\right)\left({^b\negthinspace\log}\dfrac1c\right)\left({^c\negthinspace\log}\dfrac1a\right)&=\left({^a\negthinspace\log}b^{-1}\right)\left({^b\negthinspace\log}c^{-1}\right)\left({^c\negthinspace\log}a^{-1}\right)\\ &=(-1)(-1)(-1)\left({^a\negthinspace\log}b\right)\left({^b\negthinspace\log}c\right)\left({^c\negthinspace\log}a\right)\\ &=\boxed{\boxed{-1}} \end{aligned}
Jadi, \left({^a\negthinspace\log}\dfrac1b\right)\left({^b\negthinspace\log}\dfrac1c\right)\left({^c\negthinspace\log}\dfrac1a\right)=-1.
JAWAB: E

No.

Nilai dari ^{25}\negthinspace\log\dfrac1{64}\cdot{^4\negthinspace\log}10+{^{25}\negthinspace\log}8 adalah ....
  1. -\dfrac32
  2. -\dfrac12
  3. \dfrac12
  1. 1
  2. \dfrac32
ALTERNATIF PENYELESAIAN
\begin{aligned} ^{25}\negthinspace\log\dfrac1{64}\cdot{^4\negthinspace\log}10+{^{25}\negthinspace\log}8&=^{25}\negthinspace\log4^{-3}\cdot{^4\negthinspace\log}10+{^{25}\negthinspace\log}8\\ &=-3\cdot{^{25}\negthinspace\log}4\cdot{^4\negthinspace\log}10+{^{25}\negthinspace\log}8\\ &=-3\cdot{^{25}\negthinspace\log}10+{^{25}\negthinspace\log}8\\ &={^{25}\negthinspace\log}10^{-3}+{^{25}\negthinspace\log}8\\ &={^{25}\negthinspace\log}\dfrac1{1000}+{^{25}\negthinspace\log}8\\ &={^{25}\negthinspace\log}\left(\dfrac1{1000}\cdot8\right)\\ &={^{25}\negthinspace\log}\dfrac8{1000}\\ &={^{25}\negthinspace\log}\dfrac1{125}\\ &={^{5^2}\negthinspace\log}5^{-3}\\ &=\boxed{\boxed{-\dfrac32}} \end{aligned}
Jadi, ^{25}\negthinspace\log\dfrac1{64}\cdot{^4\negthinspace\log}10+{^{25}\negthinspace\log}8=-\dfrac32.
JAWAB: A

No.

Sederhanakanlah
log 100 + log 0,1
ALTERNATIF PENYELESAIAN
\begin{aligned} \log 100+\log0,1&=\log (100\cdot0,1)\\ &=\log10\\ &=1 \end{aligned}
Jadi, log 100 + log 0,1 = 1.

No.

Sederhanakanlah ^3\negthinspace\log9-{^3\negthinspace\log}\dfrac13
ALTERNATIF PENYELESAIAN
\begin{aligned} ^3\negthinspace\log9-{^3\negthinspace\log}\dfrac13&={^3\negthinspace\log\dfrac9{\dfrac13}}\\[6pt] &={^3\negthinspace\log27}\\ &=3 \end{aligned}
Jadi, ^3\negthinspace\log9-{^3\negthinspace\log}\dfrac13=3.

No.

Sederhanakanlah
alog 1 + alog 1
ALTERNATIF PENYELESAIAN
\begin{aligned} ^a\negthinspace\log1+{^a\negthinspace\log1}&=0+0\\ &=0 \end{aligned}
Jadi, ^a\negthinspace\log1+{^a\negthinspace\log1}=0.

No.

Sederhanakanlah
^2\negthinspace\log\dfrac18+{^2\negthinspace\log\dfrac1{64}}
ALTERNATIF PENYELESAIAN
\begin{aligned} ^2\negthinspace\log\dfrac18+{^2\negthinspace\log\dfrac1{64}}&={^2\negthinspace\log\dfrac1{2^3}}+{^2\negthinspace\log\dfrac1{2^6}}\\[4pt] &={^2\negthinspace\log2^{-3}}+{^2\negthinspace\log2^{-6}}\\ &=-3+(-6)\\ &=-9 \end{aligned}
Jadi, ^2\negthinspace\log\dfrac18+{^2\negthinspace\log\dfrac1{64}}=-9.

No.

Sederhanakanlah
^2\negthinspace\log2^8+{^2\negthinspace\log\dfrac18}
ALTERNATIF PENYELESAIAN
\begin{aligned} ^2\negthinspace\log2^8+{^2\negthinspace\log\dfrac18}&=8+{^2\negthinspace\log\dfrac1{2^{-3}}}\\[3.7pt] &=8+(-3)\\ &=5 \end{aligned}
Jadi,
JAWAB:

No.

Sederhanakanlah
^2\negthinspace\log16^{\frac12}+{^2\negthinspace\log8^{\frac13}}
ALTERNATIF PENYELESAIAN
\begin{aligned} ^2\negthinspace\log16^{\frac12}+{^2\negthinspace\log8^{\frac13}}&=^2\negthinspace\log\left(2^4\right)^{\frac12}+{^2\negthinspace\log\left(2^3\right)^{\frac13}}\\[5pt] &=^2\negthinspace\log2^2+{^2\negthinspace\log2}\\ &=2+1\\ &=3 \end{aligned}
Jadi, ^2\negthinspace\log16^{\frac12}+{^2\negthinspace\log8^{\frac13}}=3.



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