Exercise Zone : Logaritma (2)

Berikut ini adalah kumpulan soal mengenai logaritma. Jika ada jawaban yang salah, mohon dikoreksi melalui komentar. Terima kasih.

Tingkat Kesulitan :

Dasar SBMPTN

• 1
• 2
• 3

---

No. 11

Sederhanakanlah
\log10^{\frac12}+\log\sqrt{10^3}

Grup Whatsapp

\(\begin{aligned} \log10^{\frac12}+\log\sqrt{10^3}&=\dfrac12+\log10^{\frac32}\\[5pt] &=\dfrac12+\dfrac32\\[5pt] &=\dfrac42\\ &=\boxed{\boxed{2}} \end{aligned}\)

Jadi, \log10^{\frac12}+\log\sqrt{10^3}=2.

---

No. 12

Sederhanakanlah
^2\negthinspace\log\sqrt2:{^2\negthinspace\log\sqrt8}

Grup Whatsapp

\(\begin{aligned} ^2\negthinspace\log\sqrt2:{^2\negthinspace\log\sqrt8}&=^2\negthinspace\log2^{\frac12}:{^2\negthinspace\log\sqrt{2^3}}\\[5pt] &=\dfrac12:{^2\negthinspace\log2^{\frac32}}\\[5pt] &=\dfrac12:\dfrac32\\[5pt] &=\dfrac12\times\dfrac23\\ &=\boxed{\boxed{\dfrac13}} \end{aligned}\)

Jadi, ^2\negthinspace\log\sqrt2:{^2\negthinspace\log\sqrt8}=\dfrac13.

---

No. 13

Sederhanakanlah
^4\negthinspace\log2+{^4\negthinspace\log32}

Grup Whatsapp

\(\begin{aligned} ^4\negthinspace\log2+{^4\negthinspace\log32}&={^{2^2}\negthinspace\log2}+{^{2^2}\negthinspace\log2^5}\\ &=\dfrac12+\dfrac52\\[5pt] &=\dfrac62\\ &=\boxed{\boxed{3}} \end{aligned}\)

5

Jadi, ^4\negthinspace\log2+{^4\negthinspace\log32}=3.

---

No. 14

Jika x\gt0 dan y\gt0, maka \dfrac{3-3\log^2xy}{1-\log x^3y^2+2\log x\sqrt{y}}= ....

UN SMA 2018

\(\begin{aligned} \dfrac{3-3\log^2xy}{1-\log x^3y^2+2\log x\sqrt{y}}&=\dfrac{3\left(1-\log^2xy\right)}{1-\left(\log x^3y^2-\log\left(x\sqrt{y}\right)^2\right)}\\ &=\dfrac{3\left(1+\log xy\right)\left(1-\log xy\right)}{1-\left(\log x^3y^2-\log x^2y\right)}\\ &=\dfrac{3\left(1+\log xy\right)\left(1-\log xy\right)}{1-\log\dfrac{x^3y^2}{x^2y}}\\ &=\dfrac{3\left(1+\log xy\right)\cancel{\left(1-\log xy\right)}}{\cancel{1-\log xy}}\\ &=3+3\log xy \end{aligned}\)

Jadi,

---

No. 15

Diketahui nilai dari ^{45}\negthinspace\log72=a dan ^{20}\negthinspace\log180=b, maka nilai dari ^3\negthinspace\log5 adalah

2. 
   
3. 
   

5. 
   

\(\begin{aligned} ^{45}\negthinspace\log72=a\\ \dfrac{^3\negthinspace\log72}{^3\negthinspace\log45}&=a\\ \dfrac{^3\negthinspace\log\left(3^2\cdot2^3\right)}{^3\negthinspace\log\left(3^2\cdot5\right)}&=a\\ \dfrac{2+3\ ^3\negthinspace\log2}{2+{^3\negthinspace\log5}}&=a\\ 2+3\ ^3\negthinspace\log2&=2a+a\ ^3\negthinspace\log5\\ 3\ ^3\negthinspace\log2-a\ ^3\negthinspace\log5&=2(a-1) \end{aligned}\)

\(\begin{aligned} ^{20}\negthinspace\log180&=b\\ \dfrac{^3\negthinspace\log180}{^3\negthinspace\log20}&=b\\ \dfrac{^3\negthinspace\log\left(3^2\cdot2^2\cdot5\right)}{^3\negthinspace\log\left(2^2\cdot5\right)}&=b\\ \dfrac{2+2\ ^3\negthinspace\log2+{^3\negthinspace\log5}}{2\ ^3\negthinspace\log2+{^3\negthinspace\log5}}&=b\\ 2+2\ ^3\negthinspace\log2+{^3\negthinspace\log5}&=2b\ ^3\negthinspace\log2+b\ ^3\negthinspace\log5\\ 2(b-1)\ ^3\negthinspace\log2+(b-1)\ ^3\negthinspace\log5&=2 \end{aligned}\)

\(\begin{aligned} 2(b-1)\ ^3\negthinspace\log2+(b-1)\ ^3\negthinspace\log5&=2\qquad&\color{red}{\times3}\\ 3\ ^3\negthinspace\log2-a\ ^3\negthinspace\log5&=2(a-1)\qquad&\color{red}{\times2(b-1)} \end{aligned}\)

\(\begin{aligned} 6(b-1)\ ^3\negthinspace\log2+3(b-1)\ ^3\negthinspace\log5&=6\\ 6(b-1)\ ^3\negthinspace\log2-2a(b-1)\ ^3\negthinspace\log5&=4(a-1)(b-1)\qquad-\\\hline (3+2a)(b-1)\ ^3\negthinspace\log5&=6-4(a-1)(b-1)\\ ^3\negthinspace\log5&=\dfrac{6-4(a-1)(b-1)}{(3+2a)(b-1)}\\ &=\boxed{\boxed{\dfrac{6-4(a-1)(b-1)}{(2a+3)(b-1)}}} \end{aligned}\)

Jadi, ^3\negthinspace\log5=\dfrac{6-4(a-1)(b-1)}{(2a+3)(b-1)}.
JAWAB: B

---

No. 16

Hasil dari \left(\dfrac{^9\negthinspace\log4\cdot{^8\negthinspace\log3}+{^3\negthinspace\log9}}{^3\negthinspace\log54-{^3\negthinspace\log2}}\right)^2=

2. 
   
3. 
   

5. 
   

\(\begin{aligned} \left(\dfrac{^9\negthinspace\log4\cdot{^8\negthinspace\log3}+{^3\negthinspace\log9}}{^3\negthinspace\log54-{^3\negthinspace\log2}}\right)^2&=\left(\dfrac{^{3^2}\negthinspace\log2^2\cdot{^{2^3}\negthinspace\log3}+2}{^3\negthinspace\log\dfrac{54}2}\right)^2\\ &=\left(\dfrac{\dfrac13\cdot{^3\negthinspace\log2}\cdot{^2\negthinspace\log3}+2}{^3\negthinspace\log27}\right)^2\\ &=\left(\dfrac{\dfrac13+2}3\right)^2\\ &=\left(\dfrac{\dfrac73}3\right)^2\\ &=\left(\dfrac79\right)^2\\ &=\boxed{\boxed{\dfrac{49}{81}}} \end{aligned}\)

Jadi, \left(\dfrac{^9\negthinspace\log4\cdot{^8\negthinspace\log3}+{^3\negthinspace\log9}}{^3\negthinspace\log54-{^3\negthinspace\log2}}\right)^2=\dfrac{49}{81}.
JAWAB: A

---

No. 17

Jika 3^a=5 dan 5^b=2 maka nilai dari {^{15}\negthinspace\log}40 adalah

1. \dfrac{2b+a}{1+a}
2. \dfrac{2ab+a}{2+a}
3. \dfrac{3ab+1}{1+a}

4. \dfrac{3ab+a}{1+a}
5. \dfrac{3b+1}{1+a}

\(\begin{aligned} 3^a&=5\\ ^3\negthinspace\log5&=a \end{aligned}\)

\(\begin{aligned} 5^b&=2\\ ^5\negthinspace\log2&=b \end{aligned}\)

\(\begin{aligned} ^3\negthinspace\log5\cdot{^5\negthinspace\log}2&=ab\\ ^3\negthinspace\log2&=ab \end{aligned}\)

\(\begin{aligned} {^{15}\negthinspace\log}40&=\dfrac{^3\negthinspace\log40}{^3\negthinspace\log15}\\[8pt] &=\dfrac{^3\negthinspace\log\left(2^3\cdot5\right)}{^3\negthinspace\log(3\cdot5)}\\[8pt] &=\dfrac{^3\negthinspace\log2^3+{^3\negthinspace\log}5}{^3\negthinspace\log3+{^3\negthinspace\log}5}\\[8pt] &=\dfrac{3\ {^3\negthinspace\log}2+{^3\negthinspace\log}5}{^3\negthinspace\log3+{^3\negthinspace\log}5}\\ &=\boxed{\boxed{\dfrac{3ab+a}{1+a}}} \end{aligned}\)

Jadi, {^{15}\negthinspace\log}40=\dfrac{3ab+a}{1+a}.
JAWAB: D

---

No. 18

Jika ^3\negmedspace\log x = 1{,}23, maka ^3\negmedspace\log 27x =

1. 3{,}690
2. -0{,}41
   
3. 3{,}23
   

4. -1{,}77
5. 4{,}23
   

Zenius.net

\(\begin{aligned} ^3\negmedspace\log 27x&={^3\negmedspace\log}27+{^3\negmedspace\log}x\\ &=3+1{,}23\\ &=\boxed{\boxed{4{,}23}} \end{aligned}\)

Jadi, ^3\negmedspace\log 27x =4{,}23.
JAWAB: E

---

No. 19

{^5\negmedspace\log 9+{^5\negmedspace\log 2}-{^5\negmedspace\log 450}=} ....

1. -2
2. -3
3. 2

4. 3
5. -1

Zenius.net

\(\begin{aligned} ^5\negmedspace\log 9+{^5\negmedspace\log 2}-{^5\negmedspace\log 450}&={^5\negmedspace\log\dfrac{9\cdot2}{450}}\\[8pt] &={^5\negmedspace\log\dfrac1{25}}\\[8pt] &={^5\negmedspace\log\dfrac1{5^2}}\\[8pt] &={^5\negmedspace\log5^{-2}}\\ &=\boxed{\boxed{-2}} \end{aligned}\)

Jadi, {^5\negmedspace\log 9+{^5\negmedspace\log 2}-{^5\negmedspace\log 450}=-2}.
JAWAB: A

---

No. 20

Hitunglah {{^2\negmedspace\log 48}-{^2\negmedspace\log 6}}

Grup Telegram

\(\eqalign{ {^2\negmedspace\log 48}-{^2\negmedspace\log 6}&={^2\negmedspace\log \dfrac{48}6}\\ &={^2\negmedspace\log 8}\\ &=\boxed{\boxed{3}} }\)

Jadi, {{^2\negmedspace\log 48}-{^2\negmedspace\log 6}=3}.

• 1
• 2
• 3