Exercise Zone : Logaritma [3]

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

Tipe:

Standar SBMPTN HOTS

No.

Tentukan {\log25\times{^2\negthinspace\log10}\times{^5\negthinspace\log4}}

Grup Telegram

ALTERNATIF PENYELESAIAN

\(\eqalign{ \log25\times{^2\negthinspace\log10}\times{^5\negthinspace\log4}&=\log25\times{^5\negthinspace\log4}\times{^2\negthinspace\log10}\\ &=\log25\times{^5\negthinspace\log2^2}\times{^2\negthinspace\log10}\\ &=\log25\times2\times{^5\negthinspace\log2}\times{^2\negthinspace\log10}\\ &=2\times\log25\times{^5\negthinspace\log10}\\ &=2\times\dfrac{\log25}{\log5}\\ &=2\times{^5\negthinspace\log25}\\ &=2\times2\\ &=\boxed{\boxed{4}} }\)

Jadi, {\log25\times{^2\negthinspace\log10}\times{^5\negthinspace\log4}=4}.

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No.

Jika {{^2\negthinspace\log3}=a}, dan {{^2\negthinspace\log5}=b}, nyatakan ^6\negthinspace\log50 dalam a dan b.

Grup Telegram

ALTERNATIF PENYELESAIAN

\(\eqalign{ ^6\negthinspace\log50&=\dfrac{^2\negthinspace\log50}{^2\negthinspace\log6}\\ &=\dfrac{^2\negthinspace\log\left(5^2\cdot2\right)}{^2\negthinspace\log(3\cdot2)}\\ &=\dfrac{{^2\negthinspace\log5^2}+{^2\negthinspace\log2}}{{^2\negthinspace\log3}+{^2\negthinspace\log2}}\\ &=\dfrac{2\ {^2\negthinspace\log5}+{^2\negthinspace\log2}}{{^2\negthinspace\log3}+{^2\negthinspace\log2}}\\ &=\boxed{\boxed{\dfrac{2b+1}{a+1}}} }\)

Jadi, ^6\negthinspace\log50=\dfrac{2b+1}{a+1}.

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No.

{{^2\negthinspace\log1}+{^3\negthinspace\log3}-{^5\negthinspace\log2}+{^5\negthinspace\log50}-{^8\negthinspace\log32}=}

Grup Telegram

ALTERNATIF PENYELESAIAN

\(\eqalign{ {^2\negthinspace\log1}+{^3\negthinspace\log3}-{^5\negthinspace\log2}+{^5\negthinspace\log50}-{^8\negthinspace\log32}&={^2\negthinspace\log1}+{^3\negthinspace\log3}+{^5\negthinspace\log50}-{^5\negthinspace\log2}-{^{2^3}\negthinspace\log{2^5}}\\ &=0+1+{^5\negthinspace\log\dfrac{50}2}-\dfrac53\\ &=1+{^5\negthinspace\log25}-\dfrac53\\ &=1+2-\dfrac53\\ &=\boxed{\boxed{\dfrac43}} }\)

Jadi, {^2\negthinspace\log1}+{^3\negthinspace\log3}-{^5\negthinspace\log2}+{^5\negthinspace\log50}-{^8\negthinspace\log32}=\dfrac43.

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No.

{{^5\negthinspace\log35}+{^5\negthinspace\log55}-{^5\negthinspace\log77}=}

Grup Telegram

ALTERNATIF PENYELESAIAN

\(\eqalign{ {^5\negthinspace\log35}+{^5\negthinspace\log55}-{^5\negthinspace\log77}&={^5\negthinspace\log\dfrac{35\cdot55}{77}}\\ &={^5\negthinspace\log25}\\ &=\boxed{\boxed{2}} }\)

Jadi, {^5\negthinspace\log35}+{^5\negthinspace\log55}-{^5\negthinspace\log77}=2.

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No.

\dfrac{a^2}{b^2}=4, maka \log\dfrac{a^3}{b^3}=

Grup Telegram

ALTERNATIF PENYELESAIAN

\(\eqalign{ \dfrac{a^2}{b^2}&=4\\ \left(\dfrac{a}b\right)^2&=4\\ \dfrac{a}b&=2 }\)

\(\eqalign{ \log\dfrac{a^3}{b^3}&=\log\left(\dfrac{a}b\right)^3\\ &=\log2^3\\ &=\boxed{\boxed{\log8}} }\)

Jadi, \log\dfrac{a^3}{b^3}=\log8.

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No.

Sederhanakan {^3\negthinspace\log\dfrac13}

Grup Telegram

ALTERNATIF PENYELESAIAN

\(\eqalign{ {^3\negthinspace\log\dfrac13}&={^3\negthinspace\log3^{-1}}\\ &=\boxed{\boxed{-1}} }\)

Jadi, {^3\negthinspace\log\dfrac13}=-1.

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No.

Jika {{^2\negthinspace\log5}=m} dan {{^2\negthinspace\log3}=n} nyatakan {^{50}\negthinspace\log90} dalam m dan n.

Grup Telegram

ALTERNATIF PENYELESAIAN

\(\eqalign{ {^{50}\negthinspace\log90}&=\dfrac{^2\negthinspace\log90}{^2\negthinspace\log50}\\ &=\dfrac{^2\negthinspace\log\left(5\cdot3^2\cdot2\right)}{^2\negthinspace\log\left(5^2\cdot2\right)}\\ &=\dfrac{{^2\negthinspace\log5}+2\ {^2\negthinspace\log3}+{^2\negthinspace\log2}}{2\ {^2\negthinspace\log5}+{^2\negthinspace\log2}}\\ &=\boxed{\boxed{\dfrac{m+2n+1}{2m+1}}} }\)

Jadi, {^{50}\negthinspace\log90}=\dfrac{m+2n+1}{2m+1}.

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No.

Nilai dari \dfrac{{^3\negthinspace\log36}\cdot{^6\negthinspace\log81}-{^4\negthinspace\log32}}{^{\frac19}\negthinspace\log27}=

1. 3
2. -9
3. \dfrac{11}2

4. -\dfrac{11}3
5. -11

Matematika Zone Id

ALTERNATIF PENYELESAIAN

\(\eqalign{ \dfrac{{^3\negthinspace\log36}\cdot{^6\negthinspace\log81}-{^4\negthinspace\log32}}{^{\frac19}\negthinspace\log27}&=\dfrac{{^3\negthinspace\log6^2}\cdot{^6\negthinspace\log81}-{^{2^2}\negthinspace\log2^5}}{^{\frac1{3^2}}\negthinspace\log3^3}\\ &=\dfrac{2\ {^3\negthinspace\log6}\cdot{^6\negthinspace\log81}-\dfrac52}{^{3^{-2}}\negthinspace\log3^3}\\ &=\dfrac{2\ {^3\negthinspace\log81}-\dfrac52}{-\dfrac32}\\ &=\dfrac{2(4)-\dfrac52}{-\dfrac32}\\ &=\dfrac{8-\dfrac52}{-\dfrac32}\\ &=\dfrac{\dfrac{11}2}{-\dfrac32}\\ &=\boxed{\boxed{-\dfrac{11}3}} }\)

Jadi, \dfrac{{^3\negthinspace\log36}\cdot{^6\negthinspace\log81}-{^4\negthinspace\log32}}{^{\frac19}\negthinspace\log27}=-\dfrac{11}3.
JAWAB: D

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No.

Hasil dari {\left(^{c^{\frac72}}\negthinspace\log b^{14}\right)\cdot\left(^{a^{\frac53}}\negthinspace\log c^{10}\right)\cdot\left(^{b^{\frac32}}\negthinspace\log a^9\right)} adalah

1. 144
2. 155
3. 166

4. 177
5. 188

SKMP Juli 2020

ALTERNATIF PENYELESAIAN

\(\eqalign{ \left(^{c^{\frac72}}\negthinspace\log b^{14}\right)\cdot\left(^{a^{\frac53}}\negthinspace\log c^{10}\right)\cdot\left(^{b^{\frac32}}\negthinspace\log a^9\right)&=\left(\dfrac27\cdot14\cdot{^c\negthinspace\log b}\right)\cdot\left(\dfrac35\cdot10\cdot{^a\negthinspace\log c}\right)\cdot\left(\dfrac23\cdot9\cdot{^b\negthinspace\log a}\right)\\ &=\left(4\cdot{^c\negthinspace\log b}\right)\cdot\left(6\cdot{^a\negthinspace\log c}\right)\cdot\left(6\cdot{^b\negthinspace\log a}\right)\\ &=144\cdot{^c\negthinspace\log b}\cdot{^a\negthinspace\log c}\cdot{^b\negthinspace\log a}\\ &=144\cdot{^a\negthinspace\log c}\cdot{^c\negthinspace\log b}\cdot{^b\negthinspace\log a}\\ &=\boxed{\boxed{144}} }\)/div>

Jadi, {\left(^{c^{\frac72}}\negthinspace\log b^{14}\right)\cdot\left(^{a^{\frac53}}\negthinspace\log c^{10}\right)\cdot\left(^{b^{\frac32}}\negthinspace\log a^9\right)}=144 .
JAWAB: A

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